@@thedoublehelix5661 i think its the other way around. in higher level math classes you have people that are comfortable with the lingo. my math teacher in 7th grade was egregious with that shit. why would you talk to a bunch on 13 year old little farts like you would talk to master students?
Maybe I should have gone to Purdue, cause it was skipped over for sure in my math classes. I'm in Calc. II and was totally confused about the notation.
The confusion with dx for me was that you can interpret dx as being the derivative of x instead of which variable to integrate. for example in the case dx/dy which interprets to be the derivative of x with respect to the variable y.
I squeaked through a year of calc by relying on short-term memory of the formulas that were fed to us way too quickly on the college quarter system. I would've actually understood it if your content existed back then. 😊
Krista, just so you know and I think you do know, Mathematics is not so hard, it has been hard for me and now I am 56 years old, just the people who teach it don't understand it in the first place because if they did, they could find easy ways to communicate it. I think you do a pretty good job, I will look at what you got in your videos. I feel I really want to master this now and know I got it finally. Thanks for your videos.
I was looking for the right way to explain this concept to students and this is the more thorough yet intuitive way I could find. I feel like even my understanding was improved by watching this video. Thank you.
Thank you so much, that was one of my biggest doubts. "dx" is one of that stuff that we just pass trought cause its not gonna make we stop the operation. You made my day, thank you !
Just discovered this channel and i love it, it does explain WHY and HOW something works, not just giving a definition(even if the clearest and easiest definition ever)
dx simply just means "a little bit of x" ∫ looks like the letter S and it means "sum" or "sum of all" now combine both and you get ∫ dx which means ”the sum of all bits of x”
Nope, it means as the change in x approaches 0. You can never get to 0, you can only approach it. It’s an idea, a concept, that you approach. Just like if you make a shape with edges and increase the edges until you approach infinity you get closer and closer to a circle. But it’s impossible to have an infinite amount of edges, hence why there is no such thing as a perfect circle. A circle is a theoretical concept you can approach.
@@SuperYtc1 wouldn’t any curve have an infinite amount of edges? And if that is the case then by your logic or what you said in the statement above, then wouldn’t that mean there is no such thing as a curve.
The best answer lies with understanding differential forms. In calc 1 and 2 (maybe 3) we like to say the dx at the end is just some notation (a reminder of what you are doing) which is integrating with respect to x. However, in calc 1 dy (or dwhatever) is typically defined as an approximation which is not the same thing that's happening when you put the dx at the end of your integral. There are a lot of really good RUclips videos which explain differential forms really well so if you really like math I suggest you search for them.
The dx notation is super clever in differential equations too imo. It lets you treat them as if they were just regular variables, making the notation much more intuitive than it originally is in calc 1, for example
After watching this video I was absolutely speechless, I mean no one has ever given me such a great explanation. I kinda help my friends, cousins, younger brothers and sisters in their math lessons, if they will ask me about dx I will directly send them to this video..... Thank u once again It was really helpful.
Wow, this video may be old, but it still helps. My teacher rushed through the explanation that the only thing I understood was limit sense from what he said, but now it's clear what we are talking about in terms of limit sense. Thank you, ma'am!
For the latter explanation in the change from delta x to dx, I think it would have been useful to make explicit that the transition comes because of the limit. As the width of the delta x becomes infinitely smaller because infinitely many (rectangles, trapezoids, squirrels [don't judge me; I regret nothing], whatever) are used to improve the approximation, the switch from delta x to dx occurs.
+The Justicar or you could just say that an integral represents the area/space/however deep you may want to go and dx represents the axis you are multiplying your function with to get the spatial dimension
The Justicar elaborate, I mean in mathemathical proofs you cant say "it symbolizes area, forget about it", but dx, delta x, delta=change in the x axis, so you are multiplying your function relative to the change of whatever axis you are integrating at the given moment, measuring the change of the function you integrate relative to the constant change of your axis.
+Ralf Anari you couldn't truthfully say that in normal conversation either. "it" in this case is "dx". And "dx" simply does not symbolize area. Further, "dx" doesn't simply represent change along the x-axis."dx" represents a specific type of change, its particular type of change only becomes precise because of the limit. "x" is the axis. "dx" represents something else. Or, if it doesn't, then you could say that dx = x. But dx doesn't equal x; d, whatever it is, must bring something else to the table that distinguishes x from dx. And what it brings is the notion of a particular type of change which is coherent only when the notion of the limit exists in conjunction with it.
The Justicar I never said it symbolizes area, it symbolizes you calculating area relative to the axis x, every integral has one constant axis and one changing axis(the function you are integrating), you average out the change in the function relative to the change of your axis and you get....wait for it....the area edit:I might have misworded it at my first comment, at the end I said dx represents teh axis, i shouldve clarified that it represents the start and the end of the points on the axis you are integrating in case you are not integrating from infinity to infinity
The integral is what the riemann sum converges to when taking the rectangles smaller and smaller. So in a way you could say that, but be careful: actually summing 0 infinitely many times is undefined, this is taking the limit which is defined
@@helloitsme7553 Actually, the more conservative way is to find largest sub-interval, find the mid-point, plug the mid point into the function, multiply them and then take the limit to find the area under the bounded region of the cuve. In three space, use the largest diagonal of the subinterval. Oh by hey! What do I know right? LOL
I have been studying origin of derivatives,and understood them completly with concepts,and from some videos i actually got it very clearly what intergration and diffrentiation really is,its not a diificult thing to understand rather people make and sound like its very hard to understand.Though i really liked your explanation for the derivative multiplying with the integral.Thanks soo much Ma'am :-)
nice explanation. most of the people are confused about it. it is a notation. in case of pure math there are many things which are confusing example when someone says let us consider the measure (X^2+1) dx. this is confusing . in case of differential forms we use the symbols dx1,dx2,.... many people are confused about this notation. here dx1 is not differential elemment of x1.
Thank you very much. I've been trying to fully understand this concept and its relation to integrals and I still find it hard to grasp. I asked a teacher bu he just told me that the dx it's an expression, which I don't agree. I understand that dy/dx = f'(x) because the change that dy experiences when dx is too small (tends to 0) is what gives the slope at any particular x value. which is the definition of a derivative. I understand that from dy/dx = f'(x) I can also write dy=f'(x) . dx. which is the formula we use to find the integral. which makes me think two things... 1) is dy the original formula then? because when I do f'(x) . dx what I find is the original formula. If so , how is that? I struggle to see it. 2) If dx is extremely small, if I multiply something for an extremely small value, wouldn't that give 0 as a result? I mean, why is it that I discard the dx when I find the primitive of f'(x)?if it's so small the result would be close to 0 so I don't understand why we just find the primite and discard the dx. This concept is driving me a bit crazy. Thank you!!
It's also worth to note that when integration occurs and you are taking an area, typically you are finding the area between two locations, so by having the dx you know the starting point, and the ending point of that specific range if the function/curve/graph.
Integrals don't always have the "dx" necessarily. In exterior/geometric calculus there are different ways of representing differential forms by observing the cross product (called wedge product) of differential basis vectors.
The last explanation was really great, but I've a question: that interpretation only applies to the definite integral? I mean, the indefinite integrals or antiderivatives aren't just the opposite of derivatives? I would appreciate your help
2. You don't need multivariable calculus in order to see why the dx is important; it already becomes apparent why it's important when one is teaching integration by substitution properly. 3. This should _always_ be mentioned _first_, since that's exactly the _historical_ reason why that notation was actually developed! All other explanations only came afterwards and are essentially unimportant.
While I agree dx is infinitely small, I think it would be appropriate to point out that it is the limit as ∆x (the width of the rectangles) approaches 0. If the function is continuous, the limit at x→a of f(x) = f(a)
I have a feeling that we had the notation straight until we started taking shortcuts with the notation, and started doing stuff like f', and stopped writing algebraically sound statements. S[] and d[] are operators. S and d cancel out to leave f-f_0. For example: using the IMPLICIT differentiation operator d[], and "=" as a binary operator: f = x^2 d[ f = x^2 ] = d[f] = 2x dx S[ d[f] = 2x dx ] = S[d[f]] = S[2x dx] = S[d[x^2]] = Sd [x^2] = x^2 - f_0, d[f_0]=0 The game is to stuff the entire expression into d[]. Because it's inside of d[], you could have added any constant the differentiates to 0. So, to say "f_0 is a constant", you add the side-condition that d[f_0]=0.
So understanding the definition of the integral (Riemann integral) you will understand why "dx" is there in the first place, when you have an limit of an infinite sum you can naturally translate it into an integral?
dx stands for INFINITESIMALLY SMALL PARTICLES OF X WHICH HAVE ACTUALLY BEEN DIFFERENTIATED THE INTEGRAL SIGN COMES FROM THE LATIN LONG S WHICH STANDS FOR SUMMA OR SHORT FOR SUMMATION Gotfried Leibnitz used these notations for the 1st time !! Hope the above helps
In Integral Calculus we do Summation of Continuous extremely small particles In Statistics we used the Capital Sigma symbol to do Summation of Discrete Variables - which may be small or even bigger - such as say 1+2 etc etc
I haven't watched any of your videos since my freshman year of college (2012). Regardless, I'm still subscribed and excited to see how much your channel has grown
Thank you. Great explanation, especially the relationship between Delta x and dx. Another reason the integral has dx is the fact that the integral represents the antiderivative. What I mean by this is, when we first learn the rules of derivatives, we are taking the derivative of a function in terms of x; it is written as an equation, and we know, "Whatever you do to one side, you must do to the other." (Another way to say this is, "However you operate on one side, you must operate the same on the other side.") So, when we take the derivative, we take the derivative of each side of the equation simultaneously: y = f(x) begets y' = f '(x). We know this also as dy/dx = f '(x), and we can multiply each side by dx to get dy = f '(x)dx. From this equation, we can now use the operation "Integral" INT[ ] on each side, giving us INT[dy] = INT[f '(x)dx]. The left side is really INT[1dy] and the antiderivative (in terms of y) is just y (let's not worry about +C at this moment). The right side antiderivative is just f(x), giving us what we started with, y = f(x) (okay ... + C). Again, thank you for this quick, important video.
Seems as though your explanation was heading toward an answer to a question I posted but not quite. I don't understand what the dx means in the front of an integral. If it means the derivative then why would we need it? Why should we care when we're actually interested in the integral? Why is it even relevant? For instance, if I want to integrate 2x, I just write x^2 + C. The dx played no role.
by infinitely summing antiderivatives you get the sum of all possible function values at every point (or sum of heights of rectangles). But to get the area under the curve you need to multuply it by the dx (which is the base of the rectangles).
THAT was AWESOME !! Thanks to. YOU i FINALLY understand the dx notation...thank YOU. SO much for avery awesome explanation i am finally in Calculus. thank. YOU. again !!!!!!
When taking the limit of the reiman sum I would just think of it as f(x) x dx or “height x width” I think dx is always there because as “n” approaches infinity, the width of delta x infinitly shrinks as the number of rectangles under the curve infinitely increase in number. Think of it as the b-a/♾
Where did you learn this from? I'm in a University of technology and they don't teach things like this, nor is it in any of the University math books that I know of. I would be very interested to know if there is a math book, which covers things like these, which are not told in regular math books or university courses
That's why I paid RUclips For Remove Ads, & That's why RUclips is Heaven for Self-Study, before watching this video i don't know why i's even using🤯amazing vid ❤
You are right, probably. But when we are supposed to find out an area under a curve, we do not count any area of each rectangle, instead of this we shoud find аntiderivative of a function. After this we subtract values, which are limits of the integration, using аntiderivative of a function. Sorry for grammatical mistakes, I am from Russia.
in measure theory we use the d(something) to notate with respect to which measure to integrate, such as dμ, dλ . This way it also does not feel redundant anymore.
I learned to think of dx as "delta of the x values" but such an infinitesimally small change in x that it can only be defined as "not no change in x." I have a BS in Mathematics and one of the best classes I ever took was a low-level class where we used no textbook and didn't do a whole lot of calculations -- we did a lot of writing. We had to write in paragraphs how we would go about solving a particular problem. It wasn't a proofs class, and we weren't using theorems. Our professor's goal was to get us to think, not to calculate. I can only imagine how much his TA hated him because each student's solution was unique. Some people would solve a problem in two paragraphs, other people took two pages; what mattered is that you arrived at the destination. Anyway, that's just a very long-winded way of me telling you that I appreciate your explanation.
arvind d Because Derivative of a function is the slope of the function at any given point. Usually on straight lines formula of slope is change in y over change in x. m=y/x or m=Δy/Δx. Since y=f(x) d/dx f(x) also means d.f(x)/dx
Search for the slope formula which is m = y2 - y1/x2-x1 so when we want the slope to be touching a single point in the curve(that means our two points to get the slope must be sooo close going to zero) we say is the difference of y / difference of x so dy/dx
If dx is only notation and it is used only to identify which term to integrate then, why we convert dx in terms of d(theta) during integration in trignometric substitution....yeahh you might say since we are integrating theta so we convert dx in terms of theta but my question is if dx is only for notation for understanding which part to integrate then dx should be completely useless but in trigonometric substitution we use dx in different way......and if dx is just notation then in trig substitution why can't we just replace dx with theta only but that's wrong...there might be big connection. we can pretty much easily recognise which terms to integrate...i think we don't need just notation........i am currently studying these minor things to fully understand mathematics but i lack people who can support me n i can't find my all questions in internet and these type of questions are neither available in text books...i hope i could take tution with a mathematician😅😁😉
Very good videos. I have some questions, what if I make online maths courses in Russian, so to what extent I can use your videos as a reference? How to I cite it? Do I have rights to use it? Thanks
Considering dx as just "integral with respect to x" is I think very misleading. Because in some integrations like u substitution, we also substitute the dx with something else in terms of u. How can we do that if it just indicates a notation? We can not cancel out the derivative sign but how can we cancel out dx? Also in higher math like differential equations we sometimes get rid of dx or dy like we do it in division. This generates even more confusion. Remove dx with something else like wording "with respect to x", then you get stuck with nearly all integrations.
So, I’ve derived sum of the rectangles for an area under the curve. But then, I stuck on this: where did we get this integral symbol, why does it represents exact area and how we went from (triangle) x to dx?
An integral is really just a sum over a continuous space. When approximating the area, you sum the rectangles like this: Σ f(x) Δx. In this expression Σ is the greek letter 'sigma' which is an 's' which stands for 'sum'. This becomes the integral symbol, which is also just an 's' but now it indicates that the sum is happening over a continuous space rather than a discrete set like the rectangles. The Δ is the greek letter 'delta' which is a 'd' and it means 'change'. This becomes the 'd' in dx and it still represents the change in x, but now because the sum is continuous, the change is infinitesimally small.
Maybe it would be even more accurate (or intuitive to some) to say "dx is projection of a section of a function curve, that is so small (the section) that the section can be viewed as a linear function, or a straight line". And then one could say "oh yeah, the dy/dx is an actual derivative of the function at that spot".
Let's say: dy/dx = 2 If I move the dx to the RHS, I get: dy = 2*dx If I integrate both sides, it will look like this: int. dy = int. 2 dx int. 1 dy = int. 2 dx dy = 1*dy If you complete the integration, you get: y = 2x + c That's what I learnt in school. Correct me if I'm wrong.
Krista, I'm looking for the cost of your service. I saw $27 flash onto the screen, but quickly disappeared. I really liked your explination of dx. I haven't looked at anything else yet.
We were told to write the dx but never told why or what it means. In an advanced course we were told to use the dx as a 'part' in integration by parts but never told why. Would be interesting to know what the dx means in integration by parts as this wasn't explained here. There appears to be more to the dx than just saying 'with respect to x'. I struggle with advanced calculus because of this and other missing information.
Essentially, integration by parts is a rearrangement of the product rule. The product rule: d/dx u*v = u*dv/dx + v*du/dx Rearrange for u*dv/dx: u*dv/dx = d/dx u*v - v*du/dx Integrate relative to x, to remove the dx from the denominator: integral u*dv = u*v - integral v*du What it means when dx is one of your parts of integration by parts, is that you are simply opting to integrate 1, so that you can differentiate the other function instead of integrating it. This way, a log or inverse trig function can be differentiated as the u-term, so that 1 dx can be the integrated term. Integral 1 dx = x.
I Think instead of dx= infinitely small distance between two x values its rather, the distance between to x values that approaches zero ( you could phrase this as a limit).
NICE. I WISH TO KNOW HOW DID YOU MAKE THIS VIDEO, I MEAN HOW CAN YOU WRITE ( e.g. by pen, which model name and brand name ) and on which device you wrote ( i mean if tablet, which device, model name, brand ) and which software to prepare video. please
but why is it necessary to add dx (the infinitesimally small distance between two super close X's) to the integration formula if it really has no effect on the integration and we're just going to end up removing it after integrating? you said it helps us find the exact area, but it has no value because its not used in the calculation and just gets ignored and removed once the integration is done, so how does it help? id appreciate an explanation from anyone
Jojo Super It tells us which variable we integrate with respect to as she said. When we use substitution and introduce a new variable( u or t) we rewrite the integral in terms of the new variable and thus dx changes to du or dt due to the substitution.
It's amazing how few math teachers actually explain things in English. 30 seconds of talking like a human can clear up years of confusion
In higher level math classes like real analysis people usually do talk like humans
Definitely
Hear! Hear!
@@thedoublehelix5661 Hej
@@thedoublehelix5661 i think its the other way around. in higher level math classes you have people that are comfortable with the lingo. my math teacher in 7th grade was egregious with that shit. why would you talk to a bunch on 13 year old little farts like you would talk to master students?
This is something that's always sort of skipped over in introductory calculus classes. Glad to see this video.
That's only true about community colleges, and that's why I attended Purdue University to learn mathematics. LOL
it's a DLC
Maybe I should have gone to Purdue, cause it was skipped over for sure in my math classes. I'm in Calc. II and was totally confused about the notation.
Yooo I was so confused about dx because it was skipped over in my calc 2 college class
@@guitarttimman fuck purdue hoosier daddy
The confusion with dx for me was that you can interpret dx as being the derivative of x instead of which variable to integrate. for example in the case dx/dy which interprets to be the derivative of x with respect to the variable y.
The d in dx stands for delta/change, so change in x / change in y = slope/rate of change
Your sound is so soothing.
It's fucking annoying, why is she whispering
Sheshan Patel voice*
Smooth change requires a smooth voice to explain it.
@@letsherptothederp Whispering is speaking without engaging your vocal cords (just using air), so she's not whispering.
@@okoyoso yeah at least to one order derivative
I made it through Calc III in college and never got an explanation for this until now. And it's so simple!
Your voice is so soothing, you need to make a math bedtime story series.
You explained this in a way that was easy to follow and pleasant to listen to. Thank you so much! New sub!
Thanks for subbing, Chek! :D
I squeaked through a year of calc by relying on short-term memory of the formulas that were fed to us way too quickly on the college quarter system. I would've actually understood it if your content existed back then. 😊
Krista, just so you know and I think you do know, Mathematics is not so hard, it has been hard for me and now I am 56 years old, just the people who teach it don't understand it in the first place because if they did, they could find easy ways to communicate it. I think you do a pretty good job, I will look at what you got in your videos. I feel I really want to master this now and know I got it finally. Thanks for your videos.
XD
dx*
hahahahahahahahahahahahhahahahah
@@sirjain4408 omg profile pic bestieeees
@@eylulvv oh yeah lol
Lol dx
@@xp-kq1im lol
I was looking for the right way to explain this concept to students and this is the more thorough yet intuitive way I could find. I feel like even my understanding was improved by watching this video. Thank you.
You're very welcome! So glad it helped! 🤗
Ohhh this is great. It's easy to find "how to"s vids, but it's these types that help integrate our understanding. Thanks again!
Thank you so much, that was one of my biggest doubts. "dx" is one of that stuff that we just pass trought cause its not gonna make we stop the operation.
You made my day, thank you !
Just discovered this channel and i love it, it does explain WHY and HOW something works, not just giving a definition(even if the clearest and easiest definition ever)
:D
I LOVE THESE VIDEOS. I USED IT FOR MY APPLIED CALC CLASS LAST YEAR AND NOW IT'S HELPING ME WITH MY ANALYTICAL ECONOMICS CLASS THIS SEMESTER. THANK YOU
+MrChocowocko You're welcome, I'm so glad the videos are helping!
At my level of study, option 2 makes the most sense for me. thank you so much, simply the best !!❤️❤️
dx simply just means "a little bit of x"
∫ looks like the letter S and it means "sum" or "sum of all"
now combine both and you get ∫ dx which means ”the sum of all bits of x”
How do u make an integral sign
ha ha that sounds hillbilly.
I hope you don't say UP AIR instead of down the street. :-)
Nope, it means as the change in x approaches 0. You can never get to 0, you can only approach it. It’s an idea, a concept, that you approach.
Just like if you make a shape with edges and increase the edges until you approach infinity you get closer and closer to a circle. But it’s impossible to have an infinite amount of edges, hence why there is no such thing as a perfect circle. A circle is a theoretical concept you can approach.
@@SuperYtc1 wouldn’t any curve have an infinite amount of edges? And if that is the case then by your logic or what you said in the statement above, then wouldn’t that mean there is no such thing as a curve.
The best answer lies with understanding differential forms. In calc 1 and 2 (maybe 3) we like to say the dx at the end is just some notation (a reminder of what you are doing) which is integrating with respect to x. However, in calc 1 dy (or dwhatever) is typically defined as an approximation which is not the same thing that's happening when you put the dx at the end of your integral. There are a lot of really good RUclips videos which explain differential forms really well so if you really like math I suggest you search for them.
The dx notation is super clever in differential equations too imo. It lets you treat them as if they were just regular variables, making the notation much more intuitive than it originally is in calc 1, for example
After watching this video I was absolutely speechless, I mean no one has ever given me such a great explanation. I kinda help my friends, cousins, younger brothers and sisters in their math lessons, if they will ask me about dx I will directly send them to this video..... Thank u once again It was really helpful.
You're welcome, I'm so glad it helped! :D
Wow, this video may be old, but it still helps. My teacher rushed through the explanation that the only thing I understood was limit sense from what he said, but now it's clear what we are talking about in terms of limit sense. Thank you, ma'am!
I have wondered this for 10 years. This was a great explanation.
Thank you, Daniel! I'm so glad it helped! :)
Congratulations on finally hitting the 100,000 subscribers! I've been waiting for weeks haha! Your videos are amazing! Keep up the great work!
+Ibrahim Awadallah Aww thank you so much! I'm so excited for the next 100K!
my teachers never teach like this way. salute u
For the latter explanation in the change from delta x to dx, I think it would have been useful to make explicit that the transition comes because of the limit. As the width of the delta x becomes infinitely smaller because infinitely many (rectangles, trapezoids, squirrels [don't judge me; I regret nothing], whatever) are used to improve the approximation, the switch from delta x to dx occurs.
+The Justicar or you could just say that an integral represents the area/space/however deep you may want to go and dx represents the axis you are multiplying your function with to get the spatial dimension
+Ralf Anari that wouldn't be so though.
The Justicar
elaborate, I mean in mathemathical proofs you cant say "it symbolizes area, forget about it", but dx, delta x, delta=change in the x axis, so you are multiplying your function relative to the change of whatever axis you are integrating at the given moment, measuring the change of the function you integrate relative to the constant change of your axis.
+Ralf Anari you couldn't truthfully say that in normal conversation either. "it" in this case is "dx". And "dx" simply does not symbolize area. Further, "dx" doesn't simply represent change along the x-axis."dx" represents a specific type of change, its particular type of change only becomes precise because of the limit.
"x" is the axis. "dx" represents something else. Or, if it doesn't, then you could say that dx = x. But dx doesn't equal x; d, whatever it is, must bring something else to the table that distinguishes x from dx. And what it brings is the notion of a particular type of change which is coherent only when the notion of the limit exists in conjunction with it.
The Justicar
I never said it symbolizes area, it symbolizes you calculating area relative to the axis x, every integral has one constant axis and one changing axis(the function you are integrating), you average out the change in the function relative to the change of your axis and you get....wait for it....the area
edit:I might have misworded it at my first comment, at the end I said dx represents teh axis, i shouldve clarified that it represents the start and the end of the points on the axis you are integrating in case you are not integrating from infinity to infinity
Integration is summation using rectangles of infinitely small width?
Not exactly, but that's the right idea.
Repeat after me: Infinitesimal. IN FA NA TESS A MUL
The integral is what the riemann sum converges to when taking the rectangles smaller and smaller. So in a way you could say that, but be careful: actually summing 0 infinitely many times is undefined, this is taking the limit which is defined
@@helloitsme7553 Actually, the more conservative way is to find largest sub-interval, find the mid-point, plug the mid point into the function, multiply them and then take the limit to find the area under the bounded region of the cuve. In three space, use the largest diagonal of the subinterval. Oh by hey! What do I know right? LOL
I have been studying origin of derivatives,and understood them completly with concepts,and from some videos i actually got it very clearly what intergration and diffrentiation really is,its not a diificult thing to understand rather people make and sound like its very hard to understand.Though i really liked your explanation for the derivative multiplying with the integral.Thanks soo much Ma'am :-)
i have studied integral for a year and still confused about what is a dx until I found your video. thanks
nice explanation. most of the people are confused about it. it is a notation. in case of pure math there are many things which are confusing example when someone says let us consider the measure (X^2+1) dx. this is confusing . in case of differential forms we use the symbols dx1,dx2,.... many people are confused about this notation. here dx1 is not differential elemment of x1.
Great explanation, very straight and rigorous at the same time. My compliments!
Thank you so much, Albert! :)
Thank you very much. I've been trying to fully understand this concept and its relation to integrals and I still find it hard to grasp. I asked a teacher bu he just told me that the dx it's an expression, which I don't agree.
I understand that dy/dx = f'(x) because the change that dy experiences when dx is too small (tends to 0) is what gives the slope at any particular x value. which is the definition of a derivative.
I understand that from dy/dx = f'(x) I can also write dy=f'(x) . dx. which is the formula we use to find the integral. which makes me think two things...
1) is dy the original formula then? because when I do f'(x) . dx what I find is the original formula. If so , how is that? I struggle to see it.
2) If dx is extremely small, if I multiply something for an extremely small value, wouldn't that give 0 as a result? I mean, why is it that I discard the dx when I find the primitive of f'(x)?if it's so small the result would be close to 0 so I don't understand why we just find the primite and discard the dx. This concept is driving me a bit crazy.
Thank you!!
Beautifully explained! As a student applying to be a physics learning assistant, I cannot wait to pass this video along to my peers.
Thank you so much, Shawn! Please do share with anyone who could use it! :D
It's also worth to note that when integration occurs and you are taking an area, typically you are finding the area between two locations, so by having the dx you know the starting point, and the ending point of that specific range if the function/curve/graph.
thank you SO much...i finally unserstand why that dx is at the end of every integration i have seen....thank you for a great demo..!!
Integrals don't always have the "dx" necessarily. In exterior/geometric calculus there are different ways of representing differential forms by observing the cross product (called wedge product) of differential basis vectors.
mechwarreir2 I'm a big fan of dropping the wedge in the wedge product, and just writing the differential forms side by side.
The last explanation was really great, but I've a question: that interpretation only applies to the definite integral?
I mean, the indefinite integrals or antiderivatives aren't just the opposite of derivatives?
I would appreciate your help
They are. The symbol was "misused" at first for the indefinite integral and people just went with it for the definite integral
2. You don't need multivariable calculus in order to see why the dx is important; it already becomes apparent why it's important when one is teaching integration by substitution properly.
3. This should _always_ be mentioned _first_, since that's exactly the _historical_ reason why that notation was actually developed! All other explanations only came afterwards and are essentially unimportant.
10 second answer, to symbolically represent the summation of infinite rectangles with area of f(x) times “dx” a small change in x.
A step closer to 100% understanding of these things. Thank you
You're very welcome, Pearl! :)
While I agree dx is infinitely small, I think it would be appropriate to point out that it is the limit as ∆x (the width of the rectangles) approaches 0. If the function is continuous, the limit at x→a of f(x) = f(a)
I have a feeling that we had the notation straight until we started taking shortcuts with the notation, and started doing stuff like f', and stopped writing algebraically sound statements. S[] and d[] are operators. S and d cancel out to leave f-f_0. For example: using the IMPLICIT differentiation operator d[], and "=" as a binary operator:
f = x^2
d[ f = x^2 ]
= d[f] = 2x dx
S[ d[f] = 2x dx ]
= S[d[f]]
= S[2x dx]
= S[d[x^2]]
= Sd [x^2]
= x^2 - f_0, d[f_0]=0
The game is to stuff the entire expression into d[]. Because it's inside of d[], you could have added any constant the differentiates to 0. So, to say "f_0 is a constant", you add the side-condition that d[f_0]=0.
So understanding the definition of the integral (Riemann integral) you will understand why "dx" is there in the first place, when you have an limit of an infinite sum you can naturally translate it into an integral?
dx stands for INFINITESIMALLY SMALL PARTICLES OF X WHICH HAVE ACTUALLY BEEN DIFFERENTIATED
THE INTEGRAL SIGN COMES FROM THE LATIN LONG S WHICH STANDS FOR SUMMA OR SHORT FOR SUMMATION
Gotfried Leibnitz used these notations for the 1st time !!
Hope the above helps
vi - "Gotfried Leibnitz used these..." Don't goof around with names!
His name is Gottfried Leibniz!
You make calculus interesting and easy to digest.
You r great !
Thank you so much, Shivam! :D
In Integral Calculus we do Summation of Continuous extremely small particles
In Statistics we used the Capital Sigma symbol to do Summation of Discrete Variables - which may be small or even bigger - such as say 1+2 etc etc
I haven't watched any of your videos since my freshman year of college (2012). Regardless, I'm still subscribed and excited to see how much your channel has grown
+Richard McGrath Wow, thank you so much Richard! I appreciate the support! :D
Thank you. Great explanation, especially the relationship between Delta x and dx. Another reason the integral has dx is the fact that the integral represents the antiderivative.
What I mean by this is, when we first learn the rules of derivatives, we are taking the derivative of a function in terms of x; it is written as an equation, and we know, "Whatever you do to one side, you must do to the other." (Another way to say this is, "However you operate on one side, you must operate the same on the other side.")
So, when we take the derivative, we take the derivative of each side of the equation simultaneously: y = f(x) begets y' = f '(x). We know this also as dy/dx = f '(x), and we can multiply each side by dx to get dy = f '(x)dx. From this equation, we can now use the operation "Integral" INT[ ] on each side, giving us INT[dy] = INT[f '(x)dx]. The left side is really INT[1dy] and the antiderivative (in terms of y) is just y (let's not worry about +C at this moment). The right side antiderivative is just f(x), giving us what we started with, y = f(x) (okay ... + C).
Again, thank you for this quick, important video.
Seems as though your explanation was heading toward an answer to a question I posted but not quite. I don't understand what the dx means in the front of an integral. If it means the derivative then why would we need it? Why should we care when we're actually interested in the integral? Why is it even relevant? For instance, if I want to integrate 2x, I just write x^2 + C. The dx played no role.
by infinitely summing antiderivatives you get the sum of all possible function values at every point (or sum of heights of rectangles). But to get the area under the curve you need to multuply it by the dx (which is the base of the rectangles).
THAT was AWESOME !! Thanks to. YOU i FINALLY understand the dx notation...thank YOU. SO much for avery awesome explanation i am finally in Calculus. thank. YOU. again !!!!!!
Thanks a lot! YT just recommended your videos to me and I love the way you explain things!
I'm in tears! You're great ma'am, keep going 👍🏻
I was just waiting for the last answer. Great video.
A trillion thanks for your time and effort! I wish you the very best Krista.
Thank you so much! :D
video stars at 0:32
Thank you
When taking the limit of the reiman sum I would just think of it as f(x) x dx or “height x width” I think dx is always there because as “n” approaches infinity, the width of delta x infinitly shrinks as the number of rectangles under the curve infinitely increase in number. Think of it as the b-a/♾
Where did you learn this from? I'm in a University of technology and they don't teach things like this, nor is it in any of the University math books that I know of. I would be very interested to know if there is a math book, which covers things like these, which are not told in regular math books or university courses
That's why I paid RUclips For Remove Ads, & That's why RUclips is Heaven for Self-Study, before watching this video i don't know why i's even using🤯amazing vid ❤
You are right, probably. But when we are supposed to find out an area under a curve, we do not count any area of each rectangle, instead of this we shoud find аntiderivative of a function. After this we subtract values, which are limits of the integration, using аntiderivative of a function. Sorry for grammatical mistakes, I am from Russia.
I wish I had a math teacher like you!
You are getting better and better, many (...or much ???) thanks from Santiago of Chile !!
Aw thanks Feria!
in measure theory we use the d(something) to notate with respect to which measure to integrate, such as dμ, dλ . This way it also does not feel redundant anymore.
I always saw the dx as a difference in x's since after integrating you had to subtract between them. You're almost to 100K subscribers by the way.
+Militant Pacifist That's a great way of thinking about it! And yes, just crossed 100K! :D
although it doesn't negatively impact you, that insight is incorrect.
Such a very wonderful explanation, yet again, Krista. Thank you very much.
You are so welcome, George! :D
very well explained ! finally got d answer to why do we actually need to integrate ! thanks !
:D
THANK YOU, this makes so much more sense now
Oh good! So glad it helped! :D
Thank you so much, Krista!
I learned to think of dx as "delta of the x values" but such an infinitesimally small change in x that it can only be defined as "not no change in x."
I have a BS in Mathematics and one of the best classes I ever took was a low-level class where we used no textbook and didn't do a whole lot of calculations -- we did a lot of writing. We had to write in paragraphs how we would go about solving a particular problem. It wasn't a proofs class, and we weren't using theorems. Our professor's goal was to get us to think, not to calculate. I can only imagine how much his TA hated him because each student's solution was unique. Some people would solve a problem in two paragraphs, other people took two pages; what mattered is that you arrived at the destination.
Anyway, that's just a very long-winded way of me telling you that I appreciate your explanation.
That sounds like a math class everybody should take! I love the idea. I'm so glad you enjoyed the video! :)
can u explain d/dx too
why division why not any other operation?
+arv synth Great idea, I'll add that to my list!
arvind d Because Derivative of a function is the slope of the function at any given point. Usually on straight lines formula of slope is change in y over change in x. m=y/x or m=Δy/Δx. Since y=f(x) d/dx f(x) also means d.f(x)/dx
thank u
It makes more sense in physics to be honest, as to why it is division. like, dd/dt = v kinda makes more sense.
Search for the slope formula which is m = y2 - y1/x2-x1 so when we want the slope to be touching a single point in the curve(that means our two points to get the slope must be sooo close going to zero) we say is the difference of y / difference of x so dy/dx
If dx is only notation and it is used only to identify which term to integrate then, why we convert dx in terms of d(theta) during integration in trignometric substitution....yeahh you might say since we are integrating theta so we convert dx in terms of theta but my question is if dx is only for notation for understanding which part to integrate then dx should be completely useless but in trigonometric substitution we use dx in different way......and if dx is just notation then in trig substitution why can't we just replace dx with theta only but that's wrong...there might be big connection. we can pretty much easily recognise which terms to integrate...i think we don't need just notation........i am currently studying these minor things to fully understand mathematics but i lack people who can support me n i can't find my all questions
in internet and these type of questions are neither available in text books...i hope i could take tution with a mathematician😅😁😉
Mejor explicación no hay! Excelente!
Very good videos. I have some questions, what if I make online maths courses in Russian, so to what extent I can use your videos as a reference? How to I cite it? Do I have rights to use it? Thanks
How do we think about the dx in an integral when the differential is viewed as a function df(x):R →R denoted df(x)(h)=y'h?
Thank you for your explanation. Now that's clear to me.
One of the best teachers in the world. I subscribed for you❤
Thank you so much for subbing! :)
This channel is so underrated i😭😭😭
Thank you so much, Nargish! :)
Thank you very much for the explanations.
Great explanation! Thanks!
good explanation!
Could u tell me what software u used for simulate the blackboard? Thank you!
I explain here: :) www.kristakingmath.com/blog/how-i-create-my-videos hope it helps!
Thank you :)
Considering dx as just "integral with respect to x" is I think very misleading. Because in some integrations like u substitution, we also substitute the dx with something else in terms of u. How can we do that if it just indicates a notation? We can not cancel out the derivative sign but how can we cancel out dx? Also in higher math like differential equations we sometimes get rid of dx or dy like we do it in division. This generates even more confusion. Remove dx with something else like wording "with respect to x", then you get stuck with nearly all integrations.
0:54 so dx is just like a full stop, and the rets of the integral equation is your sentence, and your sentence starts with a big capital S :)
thanks for your clarity
So, I’ve derived sum of the rectangles for an area under the curve. But then, I stuck on this: where did we get this integral symbol, why does it represents exact area and how we went from (triangle) x to dx?
An integral is really just a sum over a continuous space. When approximating the area, you sum the rectangles like this: Σ f(x) Δx.
In this expression Σ is the greek letter 'sigma' which is an 's' which stands for 'sum'. This becomes the integral symbol, which is also just an 's' but now it indicates that the sum is happening over a continuous space rather than a discrete set like the rectangles.
The Δ is the greek letter 'delta' which is a 'd' and it means 'change'. This becomes the 'd' in dx and it still represents the change in x, but now because the sum is continuous, the change is infinitesimally small.
thank you ı know that why we write dx but even so you told great actually these type of videos can help us to learn about that topic more clearly
I'm so glad you liked it!
Maybe it would be even more accurate (or intuitive to some) to say "dx is projection of a section of a function curve, that is so small (the section) that the section can be viewed as a linear function, or a straight line". And then one could say "oh yeah, the dy/dx is an actual derivative of the function at that spot".
Let's say:
dy/dx = 2
If I move the dx to the RHS, I get:
dy = 2*dx
If I integrate both sides, it will look like this:
int. dy = int. 2 dx
int. 1 dy = int. 2 dx
dy = 1*dy
If you complete the integration, you get:
y = 2x + c
That's what I learnt in school. Correct me if I'm wrong.
Thanks for the explanation
Thanks for explaining this ...
If we integrate dy/dx in order to get y, we first need to multiply it by dx in order to get dy, and then integrating dy will get just y.
We do. It's only half the story to say that we integrate dy/dx in order to get y.
Thank you!! You make it so easy to understand!
You're welcome, I'm so glad it helped! :D
Krista, I'm looking for the cost of your service. I saw $27 flash onto the screen, but quickly disappeared. I really liked your explination of dx. I haven't looked at anything else yet.
Hi Bill! You saw correctly. I got your email as well. :)
We were told to write the dx but never told why or what it means. In an advanced course we were told to use the dx as a 'part' in integration by parts but never told why. Would be interesting to know what the dx means in integration by parts as this wasn't explained here. There appears to be more to the dx than just saying 'with respect to x'. I struggle with advanced calculus because of this and other missing information.
Essentially, integration by parts is a rearrangement of the product rule.
The product rule:
d/dx u*v = u*dv/dx + v*du/dx
Rearrange for u*dv/dx:
u*dv/dx = d/dx u*v - v*du/dx
Integrate relative to x, to remove the dx from the denominator:
integral u*dv = u*v - integral v*du
What it means when dx is one of your parts of integration by parts, is that you are simply opting to integrate 1, so that you can differentiate the other function instead of integrating it. This way, a log or inverse trig function can be differentiated as the u-term, so that 1 dx can be the integrated term. Integral 1 dx = x.
Thanks for this very useful video
This is a great explanation. Thank you!
You're welcome, Runnermif, glad it was helpful! :)
I haven't learn calculus yet but i understand a whole concept
I’m so glad it made sense!! :)
Excellent video, I was happy at the end - thanks
You're welcome, Jay!
I Think instead of dx= infinitely small distance between two x values its rather, the distance between to x values that approaches zero ( you could phrase this as a limit).
*congratgulations you have successfully made it to my ASMR playlist*
NICE. I WISH TO KNOW HOW DID YOU MAKE THIS VIDEO, I MEAN HOW CAN YOU WRITE ( e.g. by pen, which model name and brand name ) and on which device you wrote ( i mean if tablet, which device, model name, brand ) and which software to prepare video. please
www.kristakingmath.com/blog/how-i-create-my-videos
@@kristakingmath thank you
so well explained, thank you
but why is it necessary to add dx (the infinitesimally small distance between two super close X's) to the integration formula if it really has no effect on the integration and we're just going to end up removing it after integrating? you said it helps us find the exact area, but it has no value because its not used in the calculation and just gets ignored and removed once the integration is done, so how does it help? id appreciate an explanation from anyone
Jojo Super It tells us which variable we integrate with respect to as she said. When we use substitution and introduce a new variable( u or t) we rewrite the integral in terms of the new variable and thus dx changes to du or dt due to the substitution.