If they talk like a Pikachu an talk like a Pikachu, all important properties are preserved so it's a homeomorphism, same with dy/dx. If it works like a fraction it is one
PKMNtrainer Mann No, that is not how that works. If it does not satisfy the definition of a ratio, then it is not one. And it does not satisfy the definition. Period. It is that simple.
Dear blackpenredpen in ur sense we can't also say that 8/3 is a fraction becuz we cant take 8 from 3 .dy/dx is a tiny change in y over a tiny change in x so its i think it IS a fraction since its a in physics dx/dt =V and it is a number the tiny change dx is a number infinitly small but it still a number correct me if i am not right
10 minutes summarized: Q: Is dy/dx a fraction? A: No, dy/dx is a ratio of two unlike but related things; all fractions are either quotients or ratios, but not all ratios are fractions and all quotients are fractions. Ex 1. 5 mph is a ratio of 5 miles to (per) 1 hour, it cannot be thought of as 5 miles out of 1 hour. It simply wouldn't make sense. Ex 2. Spending $75/$100 is quotient representing the idea that $75 out of $100 was spent, it cannot be thought of as the ratio $75 to $100; in this case this would be saying there are $175 total, which is wrong.
The first example you gave is wrong dude because 5 mph is a rate ( because whenever you have 1 or per in the denominator then such fraction is called rate ) not ratio because ratio has like things in the numerator and denominator as well.
You see, before today, I knew level 1 calculus and, just to understand your videos, I learnt from Product Rule, Quotient Rule, Chain Rule to Harmonic Series to Ratio Test in 3 hours. Must be the most divine part of my life. Writing, as soon as I completed Ratio Test.
Locally, dy/dx behaves like a fraction; and assuming that dx is infinitesimally small, one can formally write dy=m*dx; and because m is a constant, integrating results in y=m*x+c, which is the linearisation/first order Taylor series expansion of the function at a given point. However, one should be careful when doing so as infinetisimal calculus (treating dy/dx like a fraction) is not without contradictions
If someone looks like a Pikachu and talks like a Pikachu, then does it make him a Pikachu? Not always. It could be a Ditto. It could be a clone created by Mewtwo.
I noticed BPRD never truly justified why dy/dx is a ratio, other than just saying "it doesn't make any sense" and proclaiming that dy/dx is similar to a ratio though analogy - that is not a proof that it isn't a fraction. All fractions can be expressed as ratios, and saying dy/dx can be expressed as a ratio or that it is similar to one doesn't rule out the possibility that dy/dx is a fraction. Change my mind
I'm glad there are several people in the comments who doesn't really take it for granted. I don't believe it either, not because I know dy/dx is a fraction, but because his proof was really poor plus I have different meanings for fractions, not only the one he gave.
Anders Källberg His argument is bad indeed. But it's actually fairly trivial to understand that dy/dx is not a fraction. Why? Because d/dx is only an operator and nothing else.
What's sad about the Indian education system is that it has equipped me to evaluate complex integrals, solve hard differential equations, without actually understanding what dy and dx are.
Indian "education" is all about pretending to be competent long enough to trick a company into hiring you over someone who actually knows what they're doing.
I heard that business about dy/dx not being a fraction during my first calculus class ("you can't separate the dx from the dy!"), then saw several years of math, physics, and chemistry professors operate it exactly as if it were a fraction. Just like you did at 9:20. Had once, just once!, one of them had suggested it was a *ratio* and not a fraction, I would have understood it fine. But no one ever did, and I just sat confused the whole time. This reminds me of when you added two integrals together, and I sat flabbergasted over the fact that you just did, while I had years of math teacher tell me "it's improper to do algebra on integrals..." Yet you did and it was perfectly legitimate!
Qiller Daemon Uh, no. You're wrong. A ratio IS a fraction. By definition, they're synonymous: they are different words that have the exact same meaning. That's beyond the point. In SOME problems in physics, you can treat it as a fraction because it's an abuse of notation that works. However, in most real-physics problems, it doesn't work. It only really works in Newtonian physics.
@@angelmendez-rivera351 are you implying that newtonian physics is not "real" physics because treating dy/dx sometimes is helpful? imo if a technique works for a general problem and there is good intuition behind that technique, then it's perfectly valid.
@@aryanjoshi3342 My comment does not imply Newtonian physics is not real physics. You said "...if a technique works for a general problem and there is a good intuition behind the technique, then it's perfectly valid." I agree with you. However, treating the dy/dx as a fraction does *not* satisfy the two criteria you provided here. The technique does NOT work in a general problem, it only works in SOME problems, as I made it very clear in my reply. Also, the technique does not carry "good intuition" behind it. As I said, it is nothing more than an abuse if notation.
I didn't understand the explanation with graphs, but the part with ratios really got me. It's such a perfect way to show that even though something looks like a fraction, it isn't a fraction.
I disagree. The are many kinds of fractions: common fractions (with integer numbers), but there are also fractions with irrational numbers. The reason dy/dx is not a fraction is that it is the limit of a fraction. Limits of fractions are not fractions. Mathematically a ratio can be expressed as a fraction and in mathematics they are handled as fractions. In wikipedia a ratio is called a form of a fraction. So, a ratio is a fraction in mathematics. Saying a ratio is not a fraction is not correct. en.wikipedia.org/wiki/Fraction_(mathematics)
dy/dx is not the limit of a fraction when you define dy and dx the way BPRP did, which is, let y=f(x), take a point y1=f(x1), let m be the slope of the straight line that is tangent to y=f(x) in x=x1, take a Dx (delta x), let dx=Dx, take the y coordinate OF THAT STRAIGHT TANGENT LINE corresponding to x1+dx, let the difference between that y coordinate and y1 be dy (which is NOT equal to Dy which is based on the original function f, not its tangent line). Defined in this way, dx and dy are finite (and could be actually quite large) values that are independent of a limit. Under these conditions, it happens to be that lim| Dx->0| Dy/Dx = dy/dx (under certain conditions like f is defined, continuous and smooth in x1). So while dy and dx are finite (i.e. not infinitesimal) numbers, their quotient is equal to the quotient of the infinitesimal values of Dy and Dx (i.e. the limit of the quotient when Dx->0, in which case also Dy will ->0 and you always have a 0/0 limit)
@@hakeem4870 Bc he's not really taking about division, but rather about ratio. If you have 2 bananas, 4 pineapples and 6 apples, the question is to find relation between the amounts of these fruits, you write it as 2:4:6 i.e. 1:2:3. If you divide those you get a nonsense unrelated to your question.
So now wiKipEdiA is the arbiter of all things mathematical? Math-wikipedia is pretty good, but by no means is is complete. By definition, a _fraction_ is a part of a whole: a ratio of two integers. If you have e/pi, that is not a fraction but a quotient. 1.2/3.8 would be considered an improper fraction, but it is usually best to look at them as quotients anyway.
Dear blackpenredpen in ur sense we can't also say that 8/3 is a fraction becuz we cant take 8 from 3 .dy/dx is a tiny change in y over a tiny change in x so its i think it IS a fraction since its a in physics dx/dt =V and it is a number the tiny change dx is a number infinitly small but it still a number correct me if i am not right
Neither dy nor dy are numbers, so dy/dx is not a fraction. It's as simple as that. "tiny change" is not a mathematically valid term, "infinitesimally small number" also isn't. (At least not in standard analysis.)
If you want to continue your pizza analogy for the 2/pi as a fraction, you can say your pizza has a diameter of 1 foot, which means you have pi feet of crust. So 2/pi is taking 2 feet of the pi feet of crust.
Oh!!! Thank you for this video!! In fact I have the same difficulties in learning calculus at first when being told that dydx is not a fraction. But it works like a fraction!! Your video is so helpful to me, a graduate student who has done tons of calculus questions, but still, don’t understand the meaning of dydx.
In single variable calculus it is similar to a fraction and you can kind of treat it like one (sometimes), but it doesn't work for multivariable calculus.
if you set up rigorous foundations to work with infinitesimal quantities then it can be treated as a fraction, but it requires some work to get to that point to begin with (look up nonstandard analysis for more info)
@@angelmendez-rivera351 formally yes, but the utility of nonstandard analysis is that it rigorously puts dy and dx on an independent footing as infinitesimals meaning that any fractional manipulation of dy/dx becomes a completely valid step instead of just notational sleight of hand originally in the time of leibniz and newton (before calculus was put on a rigorous footing) this was exactly how people thought of the derivative but in order to formalise this intuition takes a lot of machinery
majmuh24 No, nonstandard analysis does not set the footing of dx and dy as infinitesimal quantities. Nonstandard analysis is only different from standard analysis in that they use hyperreal numbers instead of real numbers. d/dx is still only a linear operator in nonstandard analysis, and dx is still only a one-form in nonstandard analysis, and there is no division of one-forms in nonstandard analysis just as there is none in standard analysis. People love to bring up nonstandard analysis yet all seem to forget about Leibniz' equivalence principle. It makes me wonder whether people ACTUALLY know anything about nonstandard analysis, or if they only bring it up because they have heard about it and they think it counts as an argument. The linear operator d/dx may be defined differently in nonstandard analysis, but it remains a linear operator only still, which is the relevant point here.
@@angelmendez-rivera351 You can divide two elements of a vector space if they are a scalar multiple of each other, which is exactly the case with the one-forms df and dx.
Both are useful in different contexts. For instance if we have y=lnx, we could differentiate with respect to x as an operator on both sides of the equation (just as you would multiply both sides by a constant, for instance” d(y)/dx=d(lnx)/dx But perhaps you have a differential equation such as y’=3y/x, ignoring the fact that we could easily solve this in our heads, we could rewrite it as dy/dx=3y/x then multiply both sides by dx/3y and get dy/3y=dx/x Which allows us to integrate and solve. So both notations are useful to clarify certain situations
@Taurus Capricorn dy/dx is actually a gradient function expressing the difference in y with respect to the difference in x, that is why the use of 'd ' for the difference.
I heard it's not realistic notation but that it helps with the chain rule (dy/du × du/dx = dy/dx) Also you can connect two -1D things (null sets) with a dot.
I kind of beg to differ: if you look at the definition of the derivative in this form: f`(x) = lim deltaX -> 0 (f(x+deltaX) - f(x)) / deltaX this is in fact a fraction. The top half is the dy and the bottom half is the dx. There is division in this equation, expression or function and anywhere there is division you have a divisor, dividend, quotient and possibly a remainder there for you have a fraction. Here's an example: f(x) = x^2 f`(x) = lim deltaX -> 0 (x + deltaX)^2 - x^2) / deltaX // substitute original function = lim deltaX -> 0 (x^2 + 2x*deltaX + deltaX^2 - x^2) / deltaX // distribute using foil method = lim deltaX -> 0 (2x*deltaX + deltaX^2) / deltaX // combine like terms = lim deltaX -> 0 (deltaX*(2x + deltaX)) / deltaX // factor out a deltaX = lim deltaX -> 0 2x + deltaX // reduce the fraction since deltaX divided by deltaX is 1 = lim deltaX -> 0 2x + (0) // the above is all algebra here is where we apply the limit and this is where the calclus starts! f`(x) = 2x Now we can use the derived function f`(x) = 2x to find the slope of any given point on the original function's f(x) = x^2 curve. We know that f(x) = x^2 is a parabola and to solve for its roots we would use the quadratic equation, however, we are not interested in its roots where it intersects any of the axes. What we are interested in is the slope of a curve at a particular point. Conventional algebra can not define the slope of a curve since it is not constant throughout its graphed function or equation, unlike linear equations. Due to variations in slope; we must find the function's or equation's derivative if it exists to find the slope of a given point on the nonlinear surface! So we can take the derived equation above and use that to find the rise over run at a given point in its antiderivative. So if we have the value 3 and we evaluate that in the original function f(x) = x^2 we end up with f(3) = 3^2 = 9. This is the actual answer from performing the operation of multiplication or power to the 2nd degree. The input and output can be seen as the coordinate pair (3,9) where x=3 and y or f(x) = 9. We know that this belongs to the parabola and to find the slope at this point on the curve we can use the same exact x value in its derivative. Therefore we end up with: f`(x) = 2x = f`(3) = 2*3 = 6. So the point (3,9) on the graph of f(x) = x^2 has a slope of 6 and the derivative f`(x) = 2x has the coordinate pair (3,6). To sum it all up; if there is division you have a fraction or portion of something!
@@j.r.8176 What is d(x) supposed to mean?! Do you mean dx? Delta y / Delta x is a mean rate of change. dy / dx is a local rate of change of a function y = f(x). dy and dx on their own are _not_ rates of changes, they are differentials Where did you get the idea from that dy and dx are rates of change???
If it walks like a Pikachu, talks like a Pikachu, smells like a Pikachu, feels like a Pikachu, and is in every way indistinguishable from a Pikachu, is it a Pikachu? I think the answer is yes. By the chain rule we know dy = dy/dx × dx. This behaviour means that dy/dx acts the exact same way that fractions behave when the numerator and denominator cancel out. By enforcing the idea that dy/dx is not a fraction, we are trying to imply that dy/dx is somehow *different* to a fraction. However, the above equivalence says that is not the case, so unless you have an example where the chain rule does not apply, dy/dx is functionally equivalent to a fraction. Furthermore, the entire premise of mathematics is to find patterns and similarities between seemingly different objects when dealt with at an abstract level. Trying to endocrtinate the next generation with this nonsense is actually counter productive to developing key mathematical intuition and pattern recognition. You are just doubling the student's workload by telling them to memorise a = a × b/a *and* dy = y'(x)dx, because there's no case where it's incorrect to associate the two. Also, for your "counterexample", 20:600:1000 is just compact notation for the *three* fractions 20/600, 20/1000 and 600/1000. It is **not** equivalent to 20/600/1000 like you implied in this video; that is just blindly applying notation without any thought behind what you are doing. Everything you can do with the combined 3 way ratio can be done with the three individual fractions, but the former is just nicer notation, nothing more. The **true** thing that is happening is that the ratio operator : is commutative, but the division operator ÷ is not (because it is defined as the inverse of multiplication whereas ratios are a standalone operation, so a/b is really a × b^-1, which is clearly not the same as b/a = b × a^-1).
While it may be a "shortcut" on my part, I'd say a fraction is a number without dimensions, whereas dy/dx has the dimensions of y over x. That's not something I saw much during school in mathematics, but the notion of dimensions is important in physics (in most cases, just checking dimensions consistencies can help notice errors). As such, a fraction would be the same dimension as "1", I believe the term is to say it has no dimension, while the derivative of a position over time has the dimension of a speed. (Angles are, by definition, a ratio of two lengths, so an angle has no dimension, which can be a bit confusing).
Today, 22nd February 2024, after probably eight or so years of watching blackpenredpen, is the first time I learned that the occasional piano intro used in these videos is the introduction to the Doraemon theme.
Is Δy / Δx a fraction? They are not infinitesimals, so that is indeed a fraction. But Δy is not a "part of a whole". But I jest, I jest! Great video, as ever.
In separable differential equations, we treat dy/dx as a fraction. For example, in Newton's Law of Cooling, dT/dt = - k (T -Ta), where dT is Temperature differential, dt is time differential, k is cooling constant, T is body's Temperature and Ta is ambient (surrounding) Temperature. To solve this equation, in the first step we multiply both sides by dt and divide both sides by (T - Ta), as if dT/dt is a fraction.
@ 3:10 Blackpen redpen confuses subtraction with division. 2/pi is not 2 centimeters out of pi centimeters with the red region marking of 2cm. Reality is that 2/pi is supposed to express that it takes pi parts to make a whole (however counterintuitively you would want to understand that) and that we have only 2 of those parts. That is 1 divided pi and then times 2. This would yield a different result that what was marked in red on the stretch of line from 0 to pi. In fact 2/pi is less than 1. So he should have colored a region that is less than 1cm. Assuming the whole to be pi in the diagram is a misrepresentation.
What about 8/2? 8 is not a part of the hole either, since you can't have 8 parts of something into 2. So, thecnically, improper fractions like these are not fractions at all?
@6:40 "Sometimes a ratio is not a fraction". What? The figure you have with lengths corresponding to dy and dx clearly shows that dy/dx can be thought of as a fraction in some sense. It is technically a limit. But in that diagram standard practice is to define dy and dx as variables satisfying dy=y'*dx so that dy/dx can be thought of as a fraction for each value you give dx for a particular point on the graph. With these definitions for dy and dx, we get that dy/dx is a fraction technically. But if we don't include them and simply treat dy/dx as the derivative ( a limit), then it is not a fraction. So there is nuance, and it depends.
The thing that interests me most about math is when it is misunderstood. I once had a math teacher, and he said the same. dy/dx is not a fraction. But then,at a time when I was in school, my instructor had to solve a differential equation. Suddenly, he multiplied both sides of the equation. I raised my hand and said: “A couple of weeks ago, you said dy/dx should be considered as a symbol. Now, you're suddenly multiplying by dx on both sides of the equation. Can you explain that?” And he said something like: “Formally, you're not supposed to do that. But here it works out.” So I asked: “Can you show me an example where it doesn't work out?” And he couldn't. So I thought: “There's probably something you haven't understood correctly.” It turns out that by looking it up in “Mathematical Analysis” by Harald Bohr (the physicist Niels Bohr's older brother) from 1940, Harald Bohr proves that dy/dx can be considered as a ratio of the differential of y divided by the same differential of x. He uses the fact that a graph simply called 'x' with a slope of 1 has exactly the same differential increment on x as the function that is being examined. So, dy/dx can definitely be considered as a fraction. It's certainly correct to consider it that way.
Im a dirty physicist. obviously I use it like a fracti0on, makes life a lot easier. Thought the mathematicians at our university hate us, for using such dirty techniques.
And here I was thinking that a fraction was an equivalence class defined on a ring R with (a, b, c, d) in R (a, b) ~ (c, d) ad = bc and b, d are not zero divisors. :(
My calc III teacher at Uni was very upset of those who treated dy/dx as a ratio, because it is not! You don't divide anything... I think it is just an unfortunate (and too familiar) notation for the differential operator
I used to think it was a notation, and now, I see it as a normal fraction, but it took me years to change my mind, and it helped me. Hints : what is dx/dx ? Does d(x²)=2x.dx ? Can we write u=sin(x) => d(u) = d(sin(x)) = cos(x)d(x) ? If dy/dx=2, then is dy =2dx ? Doesn't d(5)=0 ? If you can give d(something) a meaning by itself, then why not consider d(a)/d(b) a fraction. I don't think d()/dx is a notation for the differential operator, I think d() is a notation for the differential operator. It is sad a calc 3 teacher says you don't divide anything writing dy/dx.
dy/dx is a fraction, but the meaning of dy and dx are now changed. Leibnitz used to consider dy and dx as infinitismall but now the meaning of differentials have change. According to the modern definition in calculas a differential is the principal part of the change in a function wrt to changes in independent variables
Is dy/dx more about the difference ie: first subtract, (Rise over run) which when applied for the next operation will be a fraction to determine the gradient.
d/dx (y) is a derivative of a dependable variable y with respect to a indepent variable x.It basically it.We express it by the notatiom d/dx (y). But there is another dy/dx which behave as a fraction as we saw countless of time.That dx and dy are called differentials.differentials means infinitesimal change in a variable. They both means pretty much the same in single varibale calculus.but in multibariable calculus there are some differences. One of the key differences is differentials dy/dx means the differential of two function. On the other hand derivative dy/dx means the derivative of y as a function x.
The reasoning given in the video is bad, since it uses somewhat of a bad definition for the term "fraction"/"ratio." dy/dx is not a ratio of dy and dx. Yes, dy/dx is a rate, but one must understand is that a rate is not necessarily a ratio. Historically, average rates of change were well-understood, and those are ratios. However, the notion of instantaneous rate of change remained ill-defined until the invention of calculus. What calculus teaches us is that rates are more complicated than simple ratios, and that a good understanding of them requires an understanding of linear transformations. d/dx is an operator, defined as the composition of the linear operation L0 and the difference quotient operation. If we consider a map f : U -> V, where U and V are subsets of R, then the difference quotient operator Q is such that Qf = g, where is a map g : U Cartesian T -> V, where T contains the neighborhood around 0. Qf is given by [f(x + Δ) - f(x)]/Δ = g(x, Δ). L0 is lim (Δ -> 0) [•]. As such, (L0)g = f' = df/dx. The operator D = d/dx, then, is simply D = (L0)(Q). Thus Df = f'. This is how the derivative is understood in mathematics: it is a linear operator. Yes, g is formally a quotient of two functions of its variables, but the limit of this quotient is not a quotient itself. You must remember the limit laws: a limit of quotients is a quotient of limits if and only if the dividend and divisor both converge, provided the only the dividend converges to 0, if at all. This is not the case with the limit of g. Separating the limit into a quotient of limits yields that df = 0 and dx = 0, which both causes an indeterminacy, and which yields several contradictions. What the above proves is that df/dx is not a ratio of df and dx. df and dx are not anything in this expression. The notation merely exists because it easily encodes the definition of D. And in fact, this notation is considered obsolete by many sources anyway, since in most fields of applications and theory, it is no longer used. Now, by unfortunate coincidence of bad notation, the notation df and dx does represent actual quantities that are not necessarily related to derivatives or calculus. They represent two different quantities. In integration, dx is a measure. In differential geometry, dx is a one-form. However, mathematical theory can unify this understanding. Namely, dx is the path-element of a functional co-vector of grade 1. This accounts for both its differential nature and its measure-nature. The differential operator d is defined such that it takes a zero-form, or a zero-grade, and it turns it into a one-grade. In one variable, it so happens that the operation simplifies neatly to df = (df/dx) dx, which is more coincidence than not, giving the impression that the derivative *behaves* like a ratio. However, the algebra of differentials does not contain a division operator. It is not a division algebra. Division of differentials does not exist. 1/dx does not exist as a quantity. Thus, one cannot claim (df)/(dx) = df/dx, because in fact df/dx = (d/dx)f. The notion that df/dx is ratio-like stems purely from bad usage of notation.
δτ Non-standard analysis changes nothing of what I said, thanks to the Leibniz transfer principle. In other words, analogously to how a limit of a ratio is itself generally not a ratio, the standard part function of a quotient of hyperreal numbers is not a rational function.
@@angelmendez-rivera351 Okay, dy/dx is neither a fraction in the 'one or more equal parts of a whole' sense nor is it an algebraic or irrational fraction. But it is a quotient, so the notation you bemoan as misleading is actually not, right? Besides, couldn't the definitions of ratio and fraction be suitably extended over the hyperreals? Or can only algebraic (i. e. not even transcendental) numbers form fractions and ratios?
δτ No, dy/dx is still not a quotient. The argument I presented above covers this. You and I can agree that an expression A/B is a quotient if and only if A is an element of a set, B is an element of a set, and division between elements of the two sets is well-defined. A and B could be anything, for all I care. They could be non-numbers if you them to, but only so long as division for these non-numbers can be defined. And that is the problem: there is no set of two elements of any kind for which division is defined such that the quotient of these two elements is equal to dy/dx. The notation *is still* misleading. I don't think you understand the definition of ratio. Or if you do understand it, then you don't understand the definition of dy/dx. Division for hyperreal numbers is well-defined, provided the denominator is non-zero. So this is not a question about extending the definition of "quotient" to include ratios of hyperreal numbers: the definition already does this. But this definition/extension doesn't allow dy/dx to be a ratio anymore than the simplistic definition BPRP gave. dy/dx is not a ratio of hyperreal numbers. If it were, we would never get real-valued answers for our derivatives. The most general definition of ratio you could possibly have doesn't allow dy/dx to be a ratio, because there are no two quantities you can validly and rigorously divide to obtain dy/dx. This is just a fact.
δτ Okay, to elaborate on the point about hyperreal numbers, let me be more specific. In non-standard analysis, dy/dx is defined as the standard part of Δ(ε, y)/Δ(ε, x). Since ε is a hyperreal unit, this ratio is a quotient of hyperreal numbers, which gives us a hyperreal number. The standard part then outputs a real numbers. So we have the standard part of a ratio. But the standard part of a ratio is not itself a ratio, much like how the limit of a ratio is not itself a ratio. The standard part function is a very strange function that doesn't behave like any known type of function. You can't graph it, it's not differentiable, it doesn't have simple linear, multiplicative, or distributive properties, etc. Some mathematicians describe it as almost not a function at all. So, due to all of this, we cannot say the standard part of a quotient is itself a quotient.
ooohhhhhh now I get it, I always thought multiplying both sides by dx was technically wrong but a useful shortcut cause every textbook says "HOLD ON HOLD ON YOU CAN'T DO THAT BUT DO IT ANYWAY" on that part
*_-Trap House-_* To be more precise, d is an operator indeed, but it only acts on the space of forms. f is a zero-form. df is a one-form, defined as f' dx only if f is a function of x only. Thus, by coincidence, it is true that df = f' dx, but this does not mean you can divide both sides by dx to obtain df/dx = f'. Why? Because division of differential forms does not exist.
Answer YES!!! its ftaction for two differentials dy and dx, like f/g for functions.... :)))) and dy(∆x)/dx(∆x)=f'(x) ...differentiall its linear functional..
Define the differential df(x, Δx) = f'(x)*Δx, where f' means the limit definition of the derivative. Then dx = dx(x, Δx) = Δx, so dy/dx = (f'(x)*Δx)/Δx = f'(x).
So according to you this would be wrong: y = 2x d(y) = d(2x) d(y) = 2* d(x) d(y) / d(x) = 2 And a multivariable one: z = 2x+3y d(z) = 2d(x) + 3d(y) d(z) / d(x) = 2 d(x) / d(x) + 3 d(y) / d(x) d(z) / d(x) = 2 ( 1 ) + 3 ( 0) note: zero, because y is not a func of x
I respectfully think that dy/dx, YES, it might be or is somehow a fraction in the sense the quotient represents the change of y-variable respect to x-variable. I mean, if dy = 4 for an interval of x-length of 10 between two points, the dy/dx = 4/10 = 2/5. It is an indicator how to variables can relate and how it changes. It is very dynamic measure.
dy/dx is a fraction in terms of calculus. It's the slope of tangent at a specific point on the graph while ∆y/∆x is the slope of secant that connects the two points of the graph. dy/dx is instantaneous value while ∆y/∆x is average value.
you have to FIX the notation. d[] is an implicit differentiation operator. Nest the second derivative correctly, and it works d[ d[y]/d[x] ]/d[x] = (ddy (1/dx) + dy d[1/dx] )/dx = d^2y/(dx^2) + (dy/dx) d[1/dx] = d^2y/(dx^2) - (dy/dx)(d^x/(dx^2)) note that d[d[x]]=0 would mean "x is a straight line", in the same way that d[C]=0 would mean "C is constant" note that d is an operator defined on binary ops: d[a+b] = d[a] + d[b] d[a b] = a d[b] + b d[a] d[a^b] = b a^{b-1} d[a] + ln_e[a] a^b d[b] d[log_a[b]] = ...
This is such a nice video, you'll solve people's doubts for years to come! Videos like this which are simple and solve basic questions are very necessary
I have never been a very big fan of the usual description of dy and dx. For years I thought of dy and dx as quantities that approach 0 and, for all intents and purposes, are 0. Then I got super confused when they were not treated like 0. I had to get out of that habit by asking question after question to my tutor and eventually I came to understand dy and dx not as small quantities but as elements of a slope function. Don't think of them as quantities, think of them as what is necessary to describe the slope in a general sense.
Can you do a video on how to intuit grad, div, and curl? I'm taking multivariate calc next semester and I have a rough handle on the mechanics, but I have a hard time grasping them conceptually.
I got an A in calc I and II and totally forgot that the line connecting two points of the function is called a secant line. So I learned more than I hoped just off that one fact LOL
The element of dx is approaching zero but if we make dy/dx a fraction then dx either cannot be zero or we're dealing with an undefined value. This is one way in which dy/dx is not really a fraction.
Point for clarification: A fraction happening to be one or more non-integers can, in theory, be a rational number. For example, (2pi)/pi = 2. 2 is obviously rational as it can be written as a fraction of two integers, but it has non-integer fraction representations. That's a bit of a trivial example, but if you didnt count that, then .013 would be irrational, since its rational representation is just (0.013 * 1000) / 1000, which is equally trivial.
I expected the question to mean is dy/dx a ratio, not is it a rational fraction ... I skipped parts, but I get the impression that you elided the fact that dy and dx are infinitesimals.
Simple rule of thumb : A fraction is unitless, a ratio has units. 3 seconds out of 10 seconds = 0.3 3 kilometres per hour = 3km/1hr 2 in y per 3 in x = dy/dx
@@GabrielCazorlaPersson1 Firstly it is a rule of thumb, secondly the detail of said rule of thumb requires that the units include the thing you are measuring. That is - 10mm to the left is a different 'unit' to 20mm up.
@@GabrielCazorlaPersson1 Not entirely, same as the percentage of short people and percentage of married people are different units. The comparison you make of left and up are not the same as the comparisons you can make of left and right. Left and right are on the same dimension - they have the same 'units'. Left and up are on different dimensions - they have different 'units'. Hope that clarifies your error.
9:29 - we can multiply dx on both sides. No we cannot. We dont have any rules how to work with dx, we cant do algebraic operations like multiplying or dividing by it. Ive talked about this with my match teacher at university, Ive read about this problem even at a high quality math book. In new math books, author dont even write integral f(x) dx but they write just integral f (without the dx symbol). The symbol is just symbolic thing haha which tells us how to integrate the function. This is how pure math looks at it. Of course, physicists and other (chemists) multiply the equation by dx and nothing happens, it works, I know :D But a real mathematician would say "what have you done? You cant do that." :D Ive always find funny the "fight" between physicists and mathematicians :D
saying we're simplifying by dx is language abuse it's done in a similar way but that comes from the definition of the "d" operator : df = (df/dx).dx so you can think of it as fraction simplifying when you have to do it but not too loud either
"We dont have any rules how to work with dx, we cant do algebraic operations like multiplying or dividing by it." Math is just a framework for what we can do. Nothing is set in stone except for the rules of nature. So, pure math has no foundation in science, the application of math is what funds math. This is difficult for pure math students to understand but the discipline isnt worth anything if it has no application, useless brain excerize at best. The usefulness comes once you can apply it to real problems. Generally speaking many mathematical concepts were born from observation, they came from the application of the rules of nature. This is the same with the differential quotient. We use symbols, placeholders for the concepts of math. It doesnt matter if math doesnt generally allow an operation as long as it has clear foundation in nature. Which, math only has if you take nature as the fundamentation of all mathematical concepts. Nature accepts the way we do division and multiplication of the differential quotient, even if there is no reverse of the limes defined the symbols are just symbols, who cares what concepts are behind if the rules given are reverseable? Math isnt a rulebook, its a framework and everyone may play with the framework as long as the result works with nature.
@@RafaxDRufus Because mathematician doesnt have the rules that would tell him how to operate with it, Ive said it in my previous comment. Math has its own rules, Im not a mathematician, I multiply the equation with dx all the time haha. But I know its not the most right thing to do. But it works and thats all I need.
@@tomasblovsky5871 Yeah, I just wonder as a Physics student why mathematician are so pissed off when we multiply by dx. As long as it makes sense and behave like a fraction, why should I care if the definition doesn't really allow me to do it?
Calculus teachers on the first day: dy/dx is merely a notation for derivative ok guys? When teaching integrals: oh yeah and we put this dx here just as notation too it looks cute When suddently: ok now multiply both sides by dy
Pikachu or Doraemon?
Doraemon !!!!!
ドラえもん lol
Both
dingdang
Pikachu
Dresses like Pikachu
*Uses Doraemon Song*
😂😂😂😂😂😂
dy/dx is not a third grade fraction, it is a “calculus fraction”!
dy/dx is not a fraction. It is a ratio. As an example, pi is a ratio but can't be written as a fraction.
Eric Mohs Actually, fraction and ratio are synonymous
If they talk like a Pikachu an talk like a Pikachu, all important properties are preserved so it's a homeomorphism, same with dy/dx. If it works like a fraction it is one
PKMNtrainer Mann No, that is not how that works. If it does not satisfy the definition of a ratio, then it is not one. And it does not satisfy the definition. Period. It is that simple.
@@ericm301 but π/4 is not a rational number but it is still a fraction
dy/dx isn’t a fraction... Or is it?
*VSauce Music plays*
dy/dx = y/x confirmed
Is dy/dx a fraction?
Well yes but actually no
more like X files music
@@91722854 Y/X files.
We would be better without VSauce.
Pikachu + Doraemon music
Life is good
*Inserts Jelly-Filled Donuts*
o yes, didnt notice Doraemon music 😂
Hahahha
Mak Vinci hahhahahahaa
Dear blackpenredpen in ur sense we can't also say that 8/3 is a fraction becuz we cant take 8 from 3 .dy/dx is a tiny change in y over a tiny change in x so its i think it IS a fraction since its a in physics dx/dt =V and it is a number the tiny change dx is a number infinitly small but it still a number correct me if i am not right
bprp : ofc i will not eat the whole pizza tonight
me:
He must go to bed with an empty stomach every day :((
Right? I was gonna say... I thought they sliced pizzas to make them easier to eat
My friend : dy/dx is a fraction
Me: you are not a joker .You are the entire circus.
*clown instead of joker
I saw the poll for this yesterday, and I was super excited for this video. Thanks BPRP!
10 minutes summarized:
Q: Is dy/dx a fraction?
A: No, dy/dx is a ratio of two unlike but related things; all fractions are either quotients or ratios, but not all ratios are fractions and all quotients are fractions.
Ex 1. 5 mph is a ratio of 5 miles to (per) 1 hour, it cannot be thought of as 5 miles out of 1 hour. It simply wouldn't make sense.
Ex 2. Spending $75/$100 is quotient representing the idea that $75 out of $100 was spent, it cannot be thought of as the ratio $75 to $100; in this case this would be saying there are $175 total, which is wrong.
Wow
The first example you gave is wrong dude because 5 mph is a rate ( because whenever you have 1 or per in the denominator then such fraction is called rate ) not ratio because ratio has like things in the numerator and denominator as well.
And your 2nd example is a ratio because it has same things in numerator and denominator
People before learning calculus: I have no idea why calculus is so hard, just cancel the d in d/dx to get 1/x
People after learning calculus: FML
What does FML mean?
@@createyourownfuture3840 f**k my life
@@youknowwho8925 Oh...
@@createyourownfuture3840forge mod loader
I found this way too funny
You see, before today, I knew level 1 calculus and, just to understand your videos, I learnt from Product Rule, Quotient Rule, Chain Rule to Harmonic Series to Ratio Test in 3 hours. Must be the most divine part of my life. Writing, as soon as I completed Ratio Test.
Locally, dy/dx behaves like a fraction; and assuming that dx is infinitesimally small, one can formally write dy=m*dx; and because m is a constant, integrating results in y=m*x+c, which is the linearisation/first order Taylor series expansion of the function at a given point. However, one should be careful when doing so as infinetisimal calculus (treating dy/dx like a fraction) is not without contradictions
That's what I was gonna say, that it acts as a fraction in differential equations
There aren't any contradictions though.
Ok but how in the world did you predict 1/e
That’s a famous number in problbailty.
Nice try! This poll made it to obvious that you are using witchcraft. Begone evil fiend!
@@blackpenredpen you may make a video introduce the relationship between 1/e and natural.😁😁
Is m/s a fraction
@@blackpenredpenand the Lambert W function
If someone looks like a Pikachu and talks like a Pikachu, then does it make him a Pikachu?
Not always. It could be a Ditto. It could be a clone created by Mewtwo.
the problem is "someone" so can't be a pokemon...
Depends on your definition of a Pikachu.
Well we should perhaps define "Practically Pikachu"
what about Zoroark ???
It could also really be a Smeargle.
I noticed BPRD never truly justified why dy/dx is a ratio, other than just saying "it doesn't make any sense" and proclaiming that dy/dx is similar to a ratio though analogy - that is not a proof that it isn't a fraction. All fractions can be expressed as ratios, and saying dy/dx can be expressed as a ratio or that it is similar to one doesn't rule out the possibility that dy/dx is a fraction. Change my mind
I'm glad there are several people in the comments who doesn't really take it for granted. I don't believe it either, not because I know dy/dx is a fraction, but because his proof was really poor plus I have different meanings for fractions, not only the one he gave.
@@danielfloresretamal2471 Anders KällBerg, can I hide myself here? it's dangerous outside, please please please
Anders Källberg His argument is bad indeed. But it's actually fairly trivial to understand that dy/dx is not a fraction. Why? Because d/dx is only an operator and nothing else.
@@angelmendez-rivera351 d/dx is not the same as dy/dx, dy/dx is a short form of the formal definition of a derivative
@@angelmendez-rivera351 a fraction can be seen as an operator as well
What's sad about the Indian education system is that it has equipped me to evaluate complex integrals, solve hard differential equations, without actually understanding what dy and dx are.
Rote memorization vs. intuition development
Indian "education" is all about pretending to be competent long enough to trick a company into hiring you over someone who actually knows what they're doing.
this needs to become the “send this to your crush without context” video of the math community
Done.
I heard that business about dy/dx not being a fraction during my first calculus class ("you can't separate the dx from the dy!"), then saw several years of math, physics, and chemistry professors operate it exactly as if it were a fraction. Just like you did at 9:20. Had once, just once!, one of them had suggested it was a *ratio* and not a fraction, I would have understood it fine. But no one ever did, and I just sat confused the whole time.
This reminds me of when you added two integrals together, and I sat flabbergasted over the fact that you just did, while I had years of math teacher tell me "it's improper to do algebra on integrals..." Yet you did and it was perfectly legitimate!
Qiller Daemon Uh, no. You're wrong. A ratio IS a fraction. By definition, they're synonymous: they are different words that have the exact same meaning. That's beyond the point. In SOME problems in physics, you can treat it as a fraction because it's an abuse of notation that works. However, in most real-physics problems, it doesn't work. It only really works in Newtonian physics.
@@angelmendez-rivera351 are you implying that newtonian physics is not "real" physics because treating dy/dx sometimes is helpful? imo if a technique works for a general problem and there is good intuition behind that technique, then it's perfectly valid.
@@aryanjoshi3342 My comment does not imply Newtonian physics is not real physics.
You said "...if a technique works for a general problem and there is a good intuition behind the technique, then it's perfectly valid." I agree with you. However, treating the dy/dx as a fraction does *not* satisfy the two criteria you provided here. The technique does NOT work in a general problem, it only works in SOME problems, as I made it very clear in my reply. Also, the technique does not carry "good intuition" behind it. As I said, it is nothing more than an abuse if notation.
I didn't understand the explanation with graphs, but the part with ratios really got me. It's such a perfect way to show that even though something looks like a fraction, it isn't a fraction.
That fraction explanation reminds me of the half derivative Peyam did... 1+1=2 😂😋
Hahahahaha yea! I was filming him and I couldn’t hold my laugh!!
That was a funny one lol
I disagree. The are many kinds of fractions: common fractions (with integer numbers), but there are also fractions with irrational numbers. The reason dy/dx is not a fraction is that it is the limit of a fraction. Limits of fractions are not fractions. Mathematically a ratio can be expressed as a fraction and in mathematics they are handled as fractions. In wikipedia a ratio is called a form of a fraction. So, a ratio is a fraction in mathematics. Saying a ratio is not a fraction is not correct.
en.wikipedia.org/wiki/Fraction_(mathematics)
dy/dx is not the limit of a fraction when you define dy and dx the way BPRP did, which is, let y=f(x), take a point y1=f(x1), let m be the slope of the straight line that is tangent to y=f(x) in x=x1, take a Dx (delta x), let dx=Dx, take the y coordinate OF THAT STRAIGHT TANGENT LINE corresponding to x1+dx, let the difference between that y coordinate and y1 be dy (which is NOT equal to Dy which is based on the original function f, not its tangent line). Defined in this way, dx and dy are finite (and could be actually quite large) values that are independent of a limit. Under these conditions, it happens to be that lim| Dx->0| Dy/Dx = dy/dx (under certain conditions like f is defined, continuous and smooth in x1). So while dy and dx are finite (i.e. not infinitesimal) numbers, their quotient is equal to the quotient of the infinitesimal values of Dy and Dx (i.e. the limit of the quotient when Dx->0, in which case also Dy will ->0 and you always have a 0/0 limit)
But didn't he say 1:2:3 cannot be written coherently as a fraction?
@@hakeem4870 Bc he's not really taking about division, but rather about ratio. If you have 2 bananas, 4 pineapples and 6 apples, the question is to find relation between the amounts of these fruits, you write it as 2:4:6 i.e. 1:2:3. If you divide those you get a nonsense unrelated to your question.
@@hakeem4870 I mean, you can write it as a fraction, but not in the context BPRP speaks.
So now wiKipEdiA is the arbiter of all things mathematical? Math-wikipedia is pretty good, but by no means is is complete. By definition, a _fraction_ is a part of a whole: a ratio of two integers. If you have e/pi, that is not a fraction but a quotient. 1.2/3.8 would be considered an improper fraction, but it is usually best to look at them as quotients anyway.
Dear blackpenredpen in ur sense we can't also say that 8/3 is a fraction becuz we cant take 8 from 3 .dy/dx is a tiny change in y over a tiny change in x so its i think it IS a fraction since its a in physics dx/dt =V and it is a number the tiny change dx is a number infinitly small but it still a number correct me if i am not right
*No Offence*
Physics guys use mathematics weirdly tbh XD
@@hamiltonianpathondodecahed5236 Does that include Sir Isaac Newton?
I still stand by this statement as a freshman in college.
Neither dy nor dy are numbers, so dy/dx is not a fraction. It's as simple as that.
"tiny change" is not a mathematically valid term, "infinitesimally small number" also isn't. (At least not in standard analysis.)
I'm amazed by how well you can explain stuff, keep it up
If you want to continue your pizza analogy for the 2/pi as a fraction, you can say your pizza has a diameter of 1 foot, which means you have pi feet of crust. So 2/pi is taking 2 feet of the pi feet of crust.
Well, that's the way math should be taught in schools😂😂😂
was wondering what was that poll for. now I see that my guess was correct and the poll had a purpose. great video as always!
google didn't do me well for this question, glad to see your explanation:)
Thank you!!!!
Oh!!! Thank you for this video!! In fact I have the same difficulties in learning calculus at first when being told that dydx is not a fraction. But it works like a fraction!! Your video is so helpful to me, a graduate student who has done tons of calculus questions, but still, don’t understand the meaning of dydx.
In single variable calculus it is similar to a fraction and you can kind of treat it like one (sometimes), but it doesn't work for multivariable calculus.
if you set up rigorous foundations to work with infinitesimal quantities then it can be treated as a fraction, but it requires some work to get to that point to begin with (look up nonstandard analysis for more info)
majmuh24 Even in nonstandard analysis, dy/dx is not a fraction. It is an operator acting on a vector.
@@angelmendez-rivera351 formally yes, but the utility of nonstandard analysis is that it rigorously puts dy and dx on an independent footing as infinitesimals meaning that any fractional manipulation of dy/dx becomes a completely valid step instead of just notational sleight of hand
originally in the time of leibniz and newton (before calculus was put on a rigorous footing) this was exactly how people thought of the derivative but in order to formalise this intuition takes a lot of machinery
majmuh24 No, nonstandard analysis does not set the footing of dx and dy as infinitesimal quantities. Nonstandard analysis is only different from standard analysis in that they use hyperreal numbers instead of real numbers. d/dx is still only a linear operator in nonstandard analysis, and dx is still only a one-form in nonstandard analysis, and there is no division of one-forms in nonstandard analysis just as there is none in standard analysis.
People love to bring up nonstandard analysis yet all seem to forget about Leibniz' equivalence principle. It makes me wonder whether people ACTUALLY know anything about nonstandard analysis, or if they only bring it up because they have heard about it and they think it counts as an argument. The linear operator d/dx may be defined differently in nonstandard analysis, but it remains a linear operator only still, which is the relevant point here.
@@angelmendez-rivera351 You can divide two elements of a vector space if they are a scalar multiple of each other, which is exactly the case with the one-forms df and dx.
Aram Hăvărneanu No, you cannot. Go study some differential geometry. df is not inherently a one-form, so listing it as one is incorrect.
Physicist: short answer, yes. Long answer, hell yeah
By right we should write dy/dx as d(y)/dx but for simplicity mathematicians write it as dy/dx without the brackets.
Both are useful in different contexts. For instance if we have y=lnx, we could differentiate with respect to x as an operator on both sides of the equation (just as you would multiply both sides by a constant, for instance”
d(y)/dx=d(lnx)/dx
But perhaps you have a differential equation such as y’=3y/x, ignoring the fact that we could easily solve this in our heads, we could rewrite it as
dy/dx=3y/x
then multiply both sides by dx/3y and get
dy/3y=dx/x
Which allows us to integrate and solve.
So both notations are useful to clarify certain situations
Agreed
@Taurus Capricorn dy/dx is actually a gradient function expressing the difference in y with respect to the difference in x, that is why the use of 'd ' for the difference.
I would even prefer d(y)/d(x).
@@joluju2375 (d)(y)/(d)(x)
I heard it's not realistic notation but that it helps with the chain rule
(dy/du × du/dx = dy/dx)
Also you can connect two -1D things (null sets) with a dot.
Oh, wow, I had always thought of dy/dx as a fraction - because I've always manipulated dy/dx algebraically.
For youtube 0/0 = 0.5, cause if you look at the likes bar when there are still 0 likes and 0 dislikes the blue part covers 50% of the bar
I kind of beg to differ: if you look at the definition of the derivative in this form: f`(x) = lim deltaX -> 0 (f(x+deltaX) - f(x)) / deltaX this is in fact a fraction. The top half is the dy and the bottom half is the dx. There is division in this equation, expression or function and anywhere there is division you have a divisor, dividend, quotient and possibly a remainder there for you have a fraction.
Here's an example: f(x) = x^2
f`(x) = lim deltaX -> 0 (x + deltaX)^2 - x^2) / deltaX // substitute original function
= lim deltaX -> 0 (x^2 + 2x*deltaX + deltaX^2 - x^2) / deltaX // distribute using foil method
= lim deltaX -> 0 (2x*deltaX + deltaX^2) / deltaX // combine like terms
= lim deltaX -> 0 (deltaX*(2x + deltaX)) / deltaX // factor out a deltaX
= lim deltaX -> 0 2x + deltaX // reduce the fraction since deltaX divided by deltaX is 1
= lim deltaX -> 0 2x + (0) // the above is all algebra here is where we apply the limit and this is where the calclus starts!
f`(x) = 2x
Now we can use the derived function f`(x) = 2x to find the slope of any given point on the original function's f(x) = x^2 curve. We know that f(x) = x^2 is a parabola and to solve for its roots we would use the quadratic equation, however, we are not interested in its roots where it intersects any of the axes. What we are interested in is the slope of a curve at a particular point. Conventional algebra can not define the slope of a curve since it is not constant throughout its graphed function or equation, unlike linear equations. Due to variations in slope; we must find the function's or equation's derivative if it exists to find the slope of a given point on the nonlinear surface! So we can take the derived equation above and use that to find the rise over run at a given point in its antiderivative. So if we have the value 3 and we evaluate that in the original function f(x) = x^2 we end up with f(3) = 3^2 = 9. This is the actual answer from performing the operation of multiplication or power to the 2nd degree. The input and output can be seen as the coordinate pair (3,9) where x=3 and y or f(x) = 9. We know that this belongs to the parabola and to find the slope at this point on the curve we can use the same exact x value in its derivative. Therefore we end up with: f`(x) = 2x = f`(3) = 2*3 = 6. So the point (3,9) on the graph of f(x) = x^2 has a slope of 6 and the derivative f`(x) = 2x has the coordinate pair (3,6).
To sum it all up; if there is division you have a fraction or portion of something!
I agree :)
Your error lies in thinking that the limit of a fraction has to be a fraction itself. That doesn't follow at all.
@@j.r.8176 What is d(x) supposed to mean?! Do you mean dx?
Delta y / Delta x is a mean rate of change. dy / dx is a local rate of change of a function y = f(x). dy and dx on their own are _not_ rates of changes, they are differentials
Where did you get the idea from that dy and dx are rates of change???
you are the best math teacher ever
the best Math teacher in RUclips!!!
If it walks like a Pikachu, talks like a Pikachu, smells like a Pikachu, feels like a Pikachu, and is in every way indistinguishable from a Pikachu, is it a Pikachu?
I think the answer is yes.
By the chain rule we know dy = dy/dx × dx. This behaviour means that dy/dx acts the exact same way that fractions behave when the numerator and denominator cancel out. By enforcing the idea that dy/dx is not a fraction, we are trying to imply that dy/dx is somehow *different* to a fraction. However, the above equivalence says that is not the case, so unless you have an example where the chain rule does not apply, dy/dx is functionally equivalent to a fraction.
Furthermore, the entire premise of mathematics is to find patterns and similarities between seemingly different objects when dealt with at an abstract level. Trying to endocrtinate the next generation with this nonsense is actually counter productive to developing key mathematical intuition and pattern recognition. You are just doubling the student's workload by telling them to memorise a = a × b/a *and* dy = y'(x)dx, because there's no case where it's incorrect to associate the two.
Also, for your "counterexample", 20:600:1000 is just compact notation for the *three* fractions 20/600, 20/1000 and 600/1000. It is **not** equivalent to 20/600/1000 like you implied in this video; that is just blindly applying notation without any thought behind what you are doing. Everything you can do with the combined 3 way ratio can be done with the three individual fractions, but the former is just nicer notation, nothing more. The **true** thing that is happening is that the ratio operator : is commutative, but the division operator ÷ is not (because it is defined as the inverse of multiplication whereas ratios are a standalone operation, so a/b is really a × b^-1, which is clearly not the same as b/a = b × a^-1).
There is cases where cancelling dy and dx like a fraction will result in a wrong answer in multi variable calculus
thank you. this thing has been boggling my mind for such a long time
While it may be a "shortcut" on my part, I'd say a fraction is a number without dimensions, whereas dy/dx has the dimensions of y over x. That's not something I saw much during school in mathematics, but the notion of dimensions is important in physics (in most cases, just checking dimensions consistencies can help notice errors). As such, a fraction would be the same dimension as "1", I believe the term is to say it has no dimension, while the derivative of a position over time has the dimension of a speed.
(Angles are, by definition, a ratio of two lengths, so an angle has no dimension, which can be a bit confusing).
Today, 22nd February 2024, after probably eight or so years of watching blackpenredpen, is the first time I learned that the occasional piano intro used in these videos is the introduction to the Doraemon theme.
I really like the way he writes by switching markers
Thank you for explaining! I've been doing differential equations and it helps to know that I can use dy/dx like a fraction.
Is Δy / Δx a fraction? They are not infinitesimals, so that is indeed a fraction. But Δy is not a "part of a whole".
But I jest, I jest! Great video, as ever.
LEGEND: RATIOS LOOK LIKE FRACTIONS, WORK LIKE FRACTIONS, BUT THEY ARE NOT FRACTIONS
Oh my god I thought I accidentally clicked on the wrong video when I saw the pikachu question
That creepy pikachu smiling at me and waving at me scarred me for life
After learning complex integrals in our last lesson, we will now learn fractions.
In separable differential equations, we treat dy/dx as a fraction. For example, in Newton's Law of Cooling, dT/dt = - k (T -Ta), where dT is Temperature differential, dt is time differential, k is cooling constant, T is body's Temperature and Ta is ambient (surrounding) Temperature. To solve this equation, in the first step we multiply both sides by dt and divide both sides by (T - Ta), as if dT/dt is a fraction.
QQ: What's that music that you have fade-in @1:35? I hear it used a ton over at Ants Canada and I keep trying to find it to no avail. D:
Good job on the thumbnails and titles lately
dy/dx= y/x ... fraction?? *VSauce Music plays*
Or, is it?
I was searching precisely for this comment.
@ 3:10 Blackpen redpen confuses subtraction with division. 2/pi is not 2 centimeters out of pi centimeters with the red region marking of 2cm. Reality is that 2/pi is supposed to express that it takes pi parts to make a whole (however counterintuitively you would want to understand that) and that we have only 2 of those parts. That is 1 divided pi and then times 2. This would yield a different result that what was marked in red on the stretch of line from 0 to pi. In fact 2/pi is less than 1. So he should have colored a region that is less than 1cm. Assuming the whole to be pi in the diagram is a misrepresentation.
What about 8/2? 8 is not a part of the hole either, since you can't have 8 parts of something into 2. So, thecnically, improper fractions like these are not fractions at all?
ThiagoGlady That is a part of a whole. It doesnt have to be a part smaller than the whole for it to be a part of a whole.
I don't think that matters in cases like that because that simplifies to 4 wholes
@@fluxtwee2804
Using the same logic as him, how i am supose to eat eight slices of
pizza that only have two of them?
ThiagoGlady Not gonna lie I watched the first 5 minutes so idk
@6:40 "Sometimes a ratio is not a fraction". What? The figure you have with lengths corresponding to dy and dx clearly shows that dy/dx can be thought of as a fraction in some sense. It is technically a limit. But in that diagram standard practice is to define dy and dx as variables satisfying dy=y'*dx so that dy/dx can be thought of as a fraction for each value you give dx for a particular point on the graph. With these definitions for dy and dx, we get that dy/dx is a fraction technically. But if we don't include them and simply treat dy/dx as the derivative ( a limit), then it is not a fraction. So there is nuance, and it depends.
The thing that interests me most about math is when it is misunderstood.
I once had a math teacher, and he said the same.
dy/dx is not a fraction.
But then,at a time when I was in school, my instructor had to solve a differential equation. Suddenly, he multiplied both sides of the equation. I raised my hand and said: “A couple of weeks ago, you said dy/dx should be considered as a symbol. Now, you're suddenly multiplying by dx on both sides of the equation. Can you explain that?”
And he said something like: “Formally, you're not supposed to do that. But here it works out.”
So I asked: “Can you show me an example where it doesn't work out?”
And he couldn't.
So I thought: “There's probably something you haven't understood correctly.”
It turns out that by looking it up in “Mathematical Analysis” by Harald Bohr (the physicist Niels Bohr's older brother) from 1940,
Harald Bohr proves that dy/dx can be considered as a ratio of the differential of y divided by the same differential of x. He uses the fact that a graph simply called 'x' with a slope of 1 has exactly the same differential increment on x as the function that is being examined.
So, dy/dx can definitely be considered as a fraction. It's certainly correct to consider it that way.
Im a dirty physicist. obviously I use it like a fracti0on, makes life a lot easier. Thought the mathematicians at our university hate us, for using such dirty techniques.
Can you please prove that tan 50 +tan60+tan70=tan80?
That is not true...
@@hungryfareasternslav1823 yes it is try it on a calculator it will give the same value
@@hungryfareasternslav1823 it's true in degrees not rads
That is quite remarkable indeed :O
@@bruhmoment1835 How can it be true for degrees but not for radians? Aren't they just conversions of each other?
And here I was thinking that a fraction was an equivalence class defined on a ring R with (a, b, c, d) in R (a, b) ~ (c, d) ad = bc and b, d are not zero divisors. :(
My calc III teacher at Uni was very upset of those who treated dy/dx as a ratio, because it is not! You don't divide anything... I think it is just an unfortunate (and too familiar) notation for the differential operator
I used to think it was a notation, and now, I see it as a normal fraction, but it took me years to change my mind, and it helped me. Hints : what is dx/dx ? Does d(x²)=2x.dx ? Can we write u=sin(x) => d(u) = d(sin(x)) = cos(x)d(x) ? If dy/dx=2, then is dy =2dx ? Doesn't d(5)=0 ? If you can give d(something) a meaning by itself, then why not consider d(a)/d(b) a fraction. I don't think d()/dx is a notation for the differential operator, I think d() is a notation for the differential operator. It is sad a calc 3 teacher says you don't divide anything writing dy/dx.
dy/dx is a fraction, but the meaning of dy and dx are now changed. Leibnitz used to consider dy and dx as infinitismall but now the meaning of differentials have change. According to the modern definition in calculas a differential is the principal part of the change in a function wrt to changes in independent variables
Is dy/dx more about the difference ie: first subtract, (Rise over run) which when applied for the next operation will be a fraction to determine the gradient.
Thanks Bro. You just answered my lifelong question from 9th grade
d/dx (y) is a derivative of a dependable variable y with respect to a indepent variable x.It basically it.We express it by the notatiom d/dx (y).
But there is another dy/dx which behave as a fraction as we saw countless of time.That dx and dy are called differentials.differentials means infinitesimal change in a variable.
They both means pretty much the same in single varibale calculus.but in multibariable calculus there are some differences.
One of the key differences is differentials dy/dx means the differential of two function.
On the other hand derivative dy/dx means the derivative of y as a function x.
The reasoning given in the video is bad, since it uses somewhat of a bad definition for the term "fraction"/"ratio."
dy/dx is not a ratio of dy and dx. Yes, dy/dx is a rate, but one must understand is that a rate is not necessarily a ratio. Historically, average rates of change were well-understood, and those are ratios. However, the notion of instantaneous rate of change remained ill-defined until the invention of calculus. What calculus teaches us is that rates are more complicated than simple ratios, and that a good understanding of them requires an understanding of linear transformations. d/dx is an operator, defined as the composition of the linear operation L0 and the difference quotient operation. If we consider a map f : U -> V, where U and V are subsets of R, then the difference quotient operator Q is such that Qf = g, where is a map g : U Cartesian T -> V, where T contains the neighborhood around 0. Qf is given by [f(x + Δ) - f(x)]/Δ = g(x, Δ). L0 is lim (Δ -> 0) [•]. As such, (L0)g = f' = df/dx. The operator D = d/dx, then, is simply D = (L0)(Q). Thus Df = f'. This is how the derivative is understood in mathematics: it is a linear operator. Yes, g is formally a quotient of two functions of its variables, but the limit of this quotient is not a quotient itself. You must remember the limit laws: a limit of quotients is a quotient of limits if and only if the dividend and divisor both converge, provided the only the dividend converges to 0, if at all. This is not the case with the limit of g. Separating the limit into a quotient of limits yields that df = 0 and dx = 0, which both causes an indeterminacy, and which yields several contradictions.
What the above proves is that df/dx is not a ratio of df and dx. df and dx are not anything in this expression. The notation merely exists because it easily encodes the definition of D. And in fact, this notation is considered obsolete by many sources anyway, since in most fields of applications and theory, it is no longer used.
Now, by unfortunate coincidence of bad notation, the notation df and dx does represent actual quantities that are not necessarily related to derivatives or calculus. They represent two different quantities. In integration, dx is a measure. In differential geometry, dx is a one-form. However, mathematical theory can unify this understanding. Namely, dx is the path-element of a functional co-vector of grade 1. This accounts for both its differential nature and its measure-nature. The differential operator d is defined such that it takes a zero-form, or a zero-grade, and it turns it into a one-grade. In one variable, it so happens that the operation simplifies neatly to df = (df/dx) dx, which is more coincidence than not, giving the impression that the derivative *behaves* like a ratio. However, the algebra of differentials does not contain a division operator. It is not a division algebra. Division of differentials does not exist. 1/dx does not exist as a quantity. Thus, one cannot claim (df)/(dx) = df/dx, because in fact df/dx = (d/dx)f.
The notion that df/dx is ratio-like stems purely from bad usage of notation.
What about non-standard analysis?
δτ Non-standard analysis changes nothing of what I said, thanks to the Leibniz transfer principle. In other words, analogously to how a limit of a ratio is itself generally not a ratio, the standard part function of a quotient of hyperreal numbers is not a rational function.
@@angelmendez-rivera351
Okay, dy/dx is neither a fraction in the 'one or more equal parts of a whole' sense nor is it an algebraic or irrational fraction.
But it is a quotient, so the notation you bemoan as misleading is actually not, right?
Besides, couldn't the definitions of ratio and fraction be suitably extended over the hyperreals?
Or can only algebraic (i. e. not even transcendental) numbers form fractions and ratios?
δτ No, dy/dx is still not a quotient. The argument I presented above covers this. You and I can agree that an expression A/B is a quotient if and only if A is an element of a set, B is an element of a set, and division between elements of the two sets is well-defined. A and B could be anything, for all I care. They could be non-numbers if you them to, but only so long as division for these non-numbers can be defined. And that is the problem: there is no set of two elements of any kind for which division is defined such that the quotient of these two elements is equal to dy/dx. The notation *is still* misleading.
I don't think you understand the definition of ratio. Or if you do understand it, then you don't understand the definition of dy/dx. Division for hyperreal numbers is well-defined, provided the denominator is non-zero. So this is not a question about extending the definition of "quotient" to include ratios of hyperreal numbers: the definition already does this. But this definition/extension doesn't allow dy/dx to be a ratio anymore than the simplistic definition BPRP gave. dy/dx is not a ratio of hyperreal numbers. If it were, we would never get real-valued answers for our derivatives. The most general definition of ratio you could possibly have doesn't allow dy/dx to be a ratio, because there are no two quantities you can validly and rigorously divide to obtain dy/dx. This is just a fact.
δτ Okay, to elaborate on the point about hyperreal numbers, let me be more specific. In non-standard analysis, dy/dx is defined as the standard part of Δ(ε, y)/Δ(ε, x). Since ε is a hyperreal unit, this ratio is a quotient of hyperreal numbers, which gives us a hyperreal number. The standard part then outputs a real numbers. So we have the standard part of a ratio. But the standard part of a ratio is not itself a ratio, much like how the limit of a ratio is not itself a ratio. The standard part function is a very strange function that doesn't behave like any known type of function. You can't graph it, it's not differentiable, it doesn't have simple linear, multiplicative, or distributive properties, etc. Some mathematicians describe it as almost not a function at all. So, due to all of this, we cannot say the standard part of a quotient is itself a quotient.
ooohhhhhh now I get it, I always thought multiplying both sides by dx was technically wrong but a useful shortcut cause every textbook says "HOLD ON HOLD ON YOU CAN'T DO THAT BUT DO IT ANYWAY" on that part
Ginger Stalin It is technically only a useful shortcut. dy/dx is not a fraction because d/dx is an operator
Angel Mendez-Rivera Can’t you treat d as an operator?
*_-Trap House-_* Not quite, no. The problem is that if you do, you get a contradiction, by impliying df = 0.
*_-Trap House-_* To be more precise, d is an operator indeed, but it only acts on the space of forms. f is a zero-form. df is a one-form, defined as f' dx only if f is a function of x only. Thus, by coincidence, it is true that df = f' dx, but this does not mean you can divide both sides by dx to obtain df/dx = f'. Why? Because division of differential forms does not exist.
Answer YES!!! its ftaction for two differentials dy and dx, like f/g for functions.... :)))) and dy(∆x)/dx(∆x)=f'(x) ...differentiall its linear functional..
Thank you so much I had been searching for a video that explained this well for such a long time
Define the differential df(x, Δx) = f'(x)*Δx, where f' means the limit definition of the derivative. Then dx = dx(x, Δx) = Δx, so dy/dx = (f'(x)*Δx)/Δx = f'(x).
If, for vectors, you can say that 2v/v=2, then it is the same thing with dy/dx, and then it is a fraction
When defining the rational number set, I define the denominator as a natural number, that is {1, 2, 3, ...}. So Q={m/n|m∈Z,n∈N}.
What if we define a fraction as a division problem? Is dy/dx a division problem? dy ÷ dx?
So according to you this would be wrong:
y = 2x
d(y) = d(2x)
d(y) = 2* d(x)
d(y) / d(x) = 2
And a multivariable one:
z = 2x+3y
d(z) = 2d(x) + 3d(y)
d(z) / d(x) = 2 d(x) / d(x) + 3 d(y) / d(x)
d(z) / d(x) = 2 ( 1 ) + 3 ( 0)
note: zero, because y is not a func of x
I respectfully think that dy/dx, YES, it might be or is somehow a fraction in the sense the quotient represents the change of y-variable respect to x-variable. I mean, if dy = 4 for an interval of x-length of 10 between two points, the dy/dx = 4/10 = 2/5. It is an indicator how to variables can relate and how it changes. It is very dynamic measure.
"Is dy/dx a fraction?"
Me: Is this a trick question?
Can't wait for: is ∫f(x)dx a product?
So dy/dx is just a ratio, which looks like a fraction?
dy/dx is a fraction in terms of calculus. It's the slope of tangent at a specific point on the graph while ∆y/∆x is the slope of secant that connects the two points of the graph. dy/dx is instantaneous value while ∆y/∆x is average value.
you have to FIX the notation. d[] is an implicit differentiation operator. Nest the second derivative correctly, and it works
d[ d[y]/d[x] ]/d[x]
=
(ddy (1/dx) + dy d[1/dx] )/dx
=
d^2y/(dx^2) + (dy/dx) d[1/dx]
=
d^2y/(dx^2) - (dy/dx)(d^x/(dx^2))
note that d[d[x]]=0 would mean "x is a straight line",
in the same way that d[C]=0 would mean "C is constant"
note that d is an operator defined on binary ops:
d[a+b] = d[a] + d[b]
d[a b] = a d[b] + b d[a]
d[a^b] = b a^{b-1} d[a] + ln_e[a] a^b d[b]
d[log_a[b]] = ...
This is such a nice video, you'll solve people's doubts for years to come! Videos like this which are simple and solve basic questions are very necessary
Thanks for that. I was going to ask about differential equations where you pretend dy/dx IS a fraction. Off to have my 33/ pizza now!
You don't pretend
So I was just wondering if it's a ratio or not all along, and you just answered my question.
That was beautifully explained
you are really excellent in math and the way you explain solving every problem.
if dy is smaller than dx it is a fraction... if dy is bigger than dy, it's an improper fraction.
It could not be a rational number, but it is a fraction
No, it is not a fraction.
I have never been a very big fan of the usual description of dy and dx. For years I thought of dy and dx as quantities that approach 0 and, for all intents and purposes, are 0. Then I got super confused when they were not treated like 0. I had to get out of that habit by asking question after question to my tutor and eventually I came to understand dy and dx not as small quantities but as elements of a slope function. Don't think of them as quantities, think of them as what is necessary to describe the slope in a general sense.
Depending on what school of derivatives you're on, dx/dy may be a hyperfraction instead
Can you do a video on how to intuit grad, div, and curl? I'm taking multivariate calc next semester and I have a rough handle on the mechanics, but I have a hard time grasping them conceptually.
Try out 3blue1brown's series on the vector calculus, it's pretty good ;)
With physics everything becomes clearer
Grad? Just study mechanics
Div and curl ? E & M should give you a cool understanding of those concepts
I got an A in calc I and II and totally forgot that the line connecting two points of the function is called a secant line. So I learned more than I hoped just off that one fact LOL
The element of dx is approaching zero but if we make dy/dx a fraction then dx either cannot be zero or we're dealing with an undefined value. This is one way in which dy/dx is not really a fraction.
Your explanation is amazing notwithstanding your English (I'm trying to figure it out btw)😅😅
Love your videos and feeling so helpful.
Point for clarification: A fraction happening to be one or more non-integers can, in theory, be a rational number. For example, (2pi)/pi = 2. 2 is obviously rational as it can be written as a fraction of two integers, but it has non-integer fraction representations. That's a bit of a trivial example, but if you didnt count that, then .013 would be irrational, since its rational representation is just (0.013 * 1000) / 1000, which is equally trivial.
I wonder if it would be better to teach it as dy:dx instead of dy/dx, since that would technically be the correct formal notation for a ratio.
I expected the question to mean is dy/dx a ratio, not is it a rational fraction ... I skipped parts, but I get the impression that you elided the fact that dy and dx are infinitesimals.
Simple rule of thumb : A fraction is unitless, a ratio has units.
3 seconds out of 10 seconds = 0.3
3 kilometres per hour = 3km/1hr
2 in y per 3 in x = dy/dx
@@GabrielCazorlaPersson1 Firstly it is a rule of thumb, secondly the detail of said rule of thumb requires that the units include the thing you are measuring.
That is - 10mm to the left is a different 'unit' to 20mm up.
@@GabrielCazorlaPersson1 Not entirely, same as the percentage of short people and percentage of married people are different units.
The comparison you make of left and up are not the same as the comparisons you can make of left and right.
Left and right are on the same dimension - they have the same 'units'. Left and up are on different dimensions - they have different 'units'.
Hope that clarifies your error.
@@GabrielCazorlaPersson1 You might be right about the point, but this doesn't allow you to act as contemptuous as you do.
9:29 - we can multiply dx on both sides. No we cannot. We dont have any rules how to work with dx, we cant do algebraic operations like multiplying or dividing by it. Ive talked about this with my match teacher at university, Ive read about this problem even at a high quality math book. In new math books, author dont even write integral f(x) dx but they write just integral f (without the dx symbol). The symbol is just symbolic thing haha which tells us how to integrate the function. This is how pure math looks at it. Of course, physicists and other (chemists) multiply the equation by dx and nothing happens, it works, I know :D But a real mathematician would say "what have you done? You cant do that." :D Ive always find funny the "fight" between physicists and mathematicians :D
But why wouldn't you do that as a mathematician if it works and it acts like a fraction?
saying we're simplifying by dx is language abuse
it's done in a similar way but that comes from the definition of the "d" operator : df = (df/dx).dx
so you can think of it as fraction simplifying when you have to do it but not too loud either
"We dont have any rules how to work with dx, we cant do algebraic operations like multiplying or dividing by it."
Math is just a framework for what we can do. Nothing is set in stone except for the rules of nature. So, pure math has no foundation in science, the application of math is what funds math.
This is difficult for pure math students to understand but the discipline isnt worth anything if it has no application, useless brain excerize at best. The usefulness comes once you can apply it to real problems.
Generally speaking many mathematical concepts were born from observation, they came from the application of the rules of nature. This is the same with the differential quotient. We use symbols, placeholders for the concepts of math. It doesnt matter if math doesnt generally allow an operation as long as it has clear foundation in nature. Which, math only has if you take nature as the fundamentation of all mathematical concepts.
Nature accepts the way we do division and multiplication of the differential quotient, even if there is no reverse of the limes defined the symbols are just symbols, who cares what concepts are behind if the rules given are reverseable?
Math isnt a rulebook, its a framework and everyone may play with the framework as long as the result works with nature.
@@RafaxDRufus Because mathematician doesnt have the rules that would tell him how to operate with it, Ive said it in my previous comment. Math has its own rules, Im not a mathematician, I multiply the equation with dx all the time haha. But I know its not the most right thing to do. But it works and thats all I need.
@@tomasblovsky5871 Yeah, I just wonder as a Physics student why mathematician are so pissed off when we multiply by dx. As long as it makes sense and behave like a fraction, why should I care if the definition doesn't really allow me to do it?
Excellent explanation!
Todo un genio hermano! Saludos
Calculus teachers on the first day: dy/dx is merely a notation for derivative ok guys?
When teaching integrals: oh yeah and we put this dx here just as notation too it looks cute
When suddently: ok now multiply both sides by dy
BlackpikachuandRedpikachu???