Understanding Differentiation Part 1: The Slope of a Tangent Line
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- Опубликовано: 14 дек 2024
- The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
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Thanks so much for taking the time to explain through a comprehensive course. Great presentation and visuals have really helped me understand!
Excellent explanation. If you can believe, I have been looking for this kind of explanation for 3 decades
Your name is Rajesh and you are an Indian. Needless to say I'm an Indian too (my name) and I completely get your point. Teachers here just don't deliver the backstory, instead, plunging straight into formulas. Sad. Gracefully, I'm less unfortunate because it took me only 6 years to stumble across this vivid explanation. Cheers, Davey!
This is so incredibly helpful for me! I'm on my first semester of Chemical Engineering, and my math is, to say the very least, lacking. Professor Dave's classes on top of my Teacher's is just that extremely needed extra push. Reviewing stuff, taking formulas... I always come back to this channel for both!
And it's not just calculus! Chemistry (even the lab stuff), physics... There's everything you need and with awesome explanations! Personally, even the way Prof. Dave talks (in a non rushed, clear way) is very helpful, considering English is not my first language.
Thank you so much for what you do, professor! You're honestly so good at what you do!!
I also want to study chemical engineering. How is university?
@@ibrahimjaiteh42 Are you currently studying Engineering in university?
Man I m an engineer, but after watching your videos, I believe that I had a myth that I am engineer. Tx a million for this initiative, it's nothing short of a boon
I like your way of explaining it. I had calculus some 56 years ago and was mediocre. It’s nice to revisit.
Wow you are a very old person! Wonderful to know of such people
@@harisaisenthilkumar2880 chill
@@Daltnation no, you chill
@@harisaisenthilkumar2880 true in India it's too uncommon 😲
@@Jayyjaayy Yea bro
you are one of the best teacher I have studied from because your videos are not complicated and you explain in balanced pace😄 Thanks for all of your videos 😃
I love these videos! Never stop!
This video is better than anything out there
Next: how to play professor dave theme on piano
@Mathias Fernandes lol😂
@@mranonymous_25 s
I have looked at textbooks, I have looked at websites, and I was still stuck on finding the equation of a tangent using differentiation. Until I found this video. Subbed
Who are you? You don't seem to be an ordinary person.. you're amazing and extraordinary 🙏🤝
He is a guru
he's Jesus.
@@rosepierce9382Definitely a more plausible candidate than the orange tumor.
@@diegoarmando5489 the orange tumor? you mean the guru?
Professor, I respect everything about you and you deserve the ultimate recognition!
I had a grade 12 teacher who taught this very well, but 34 years later, I'm doing another degree in the sciences part-time. I probably won't need calculus for a little while, but it's never too soon to review. I think the problem with some first and second year math and science classes is that they try to fit too much material in too fast, so students don't absorb all the material at the beginning.
_Today's students are so lucky_
MIZORAM - Mafaka Hnamte Internet is heaven!
indeed i totally agree with you
Yep, professor dave is here to save me at all times!
But these students take it for granted
But we have lot to learn than you did so just stay silent
Thanks!
Thank you sir for your dedication and for making this free!
So to study calculus , we should know algebra and trigonometry and polynomial
It seems like it, eh? :)
What is polynomial?
Algebra is necessary for cal
@@pinklady7184 if you don’t know what a polynomial is, then don’t be here.
Skillertrap101 X lolol, my ancient comment above is a year old. *VERY OLD* comment. I have since been self-studying mathematics and I have pulled up much. I am now reading a book "Calculus, Metric Version" by James Stewart. After that, I will level up to calculus II and III. I am also reading books on real analysis, and linear algebra. I will shortly move onto topology and abstract algebra.
You are marvelous Prof Dave!
Alright so we're the pre requisites of Calculus are
> Functions
▪Exponential Functions
▪Logarithmic Functions
▪Equations of Straight Lines
> Coordinate Geometry
▪Slope Formula
> Trigonometric Identities
> Basic Geometry
> Algebra
▪Polynomials
▪Quadratic Equations
▪Binomial Theorem
> Number Theory
▪Number Systems
▪Complex Numbers
Correct me if I missed anything
one of the most succinct and clearest explanations
professor you deserve more subscribers!!
i agree! tell your friends :)
what a reply !! 😂😂😂😂👍🏻 well played !
@@ProfessorDaveExplains
Hola
He really does
Great video man! Thank you for explaining all these things to us!
Iam waiting for this many days
Which character is ur avatar man?
Onedera?
Hate to be that person but it is I * was or I * have been,
excellent way of explaining. I got amazed with your explanation and did lot of calculus sums. Sir I will be sending very special 4 questions through email to you. Please be kind enough to reply. Interestingly waiting for your explanation . Thanks
For the record: I understand he was explaining the relationship of limits of infinity and calculus, but this is how I would go about this problem in real life.
I find full simplification easier to understand, so this is what I noticed: expand the binomial on the numerator by turning x²-1 into (x+1)(x-1) and then divide the numerator and the denominator by x-1; this leaves you with m=x+1.
Adding one to the x-value and getting m=2 could have been a simpler way to approach this, although I know this specific problem was not the scope of the video.
Just here in case anyone wants to take advantage of the info. 😊
This is episode 1, slow down buddy.
@@AustinMulka 😂😂
Hey, Miguel! Can you tell me why you're adding 1 to the x-value?
@@rosepierce9382 Because if x is 1 then x-1 is zero. Then we can state that, as x-1 approaches zero, the m value approaches 2.
thanks for your videos! i'm gonna tell my friends about this channel
✨Thank you, Sir, 🙏🌺✨you are such a beautiful Teacher🙏🏼🌺✨
Sir please make video series on complete topics of differential equations 😊😊♥️♥️
Excelent explanation profesor very clear thanks
... in short ... if there is a function f: x -> y ... there MAY be a function ... f ': dx -> dy ... if so ... f is integral of f ' ... and ... f ' is derivative of f ... the logic behind is simple ... if f is bijective function of each POINT x to y ... then there MAY be bijective function f ' of each SEGMENT dx to dy ... so ... f is point to point function ... f ' is segment to segment function derived from function f ... in the infinity of arbitrary small segments it becomes differential value ... in Leibniz notation known as "d" ...
Superb sir I love listening to you
really i agree with you today's students are indeed very lucky
Yeah, we didn’t get during our high school days
extremely good job
Subscribed 🎉
You are great sir!
2:19 but why we are moving the second point professor please explain
to get closer to the first point
2:10 Couldn't we just take the first derivative of y = x^2? Then we'd get 2x, which we can take the value of m to be 2, then we plug in the points and get the same answer Y = 2x - 1. It kinda gets the job done pretty quick don't you think?
that doesn't really explain differentiation though does it....this way at least shows the concept of approximating when the change becomes infinitesimally small and how you can actually justify getting the gradient from only one point on a graph which logically doesn't really make much sense once you think about it (how can you measure "change" if you only measure one point)
Quality education!
thanks so much sir
thanks dave!
Great video however consider using an example other than one. It is hard for beginners to see exactly what is happening with one - one squared is one so it's obscure. Just a suggestion feel free to ignore.
Good
Bless you
Thanks for this nice and useful explanation,
What program is been used to make this video?
Thank you for answering me
adobe after effects
@@ProfessorDaveExplains thanks
How we initially got the basic line equation.... Parabola equation etc.... Can u make videos on how various equations have been formed .. Example : how y=mx+c for straight line...
i have over 100 math tutorials before this one in the mathematics playlist that explain everything about lines, conic sections, etc. check them out!
Professor Dave Explains thank you sir... I will def go through it...
Hey professor make videos about human physiology ur amazing
anatomy and physiology is coming very soon!
Jesus teaches Calculus. I love it.
4:25, how did he figure out that b equals to -1?? Please someone, answer my question
I think I just figured it out. I didn't notice that he plugged in the (1,1) into the equation
could you make a video series on how to do differential equations?
that will be coming soon!
I watched it.
Nice
0:49
professor, how do you make those videos? do you have app to make those animation
adobe after effects
And to be theoretical physicist we need to know calculus algebra and trigonometry and geometry
probably more than calculus, but get calculus down first!
Sir nice.
Done.
2:50 professor why you are taking point only less than 2 not above 2
because the point is to get closer and closer to the first point
Hmmm... Professor Dave, I think you have made a mistake. Isn't the answer 2x, instead of 2x-1? d/dx(x^2)... so you bring the 2 down making it 2x, and then substract a power, so 2x^(2-1) = 2x^1 = 2x. Please let me know if it is I who has made a mistake, or if the mistake is yours. Sorry if I sound harsh, I definitely don't mean to :)
The answer is 2x-1 because that's the equation of the line which was the thing we needed to find, if you just differentiate the curve you will just get the gradient function for the curve which will be 2x.
so the derivative of x2 is 2 or 2x?
how's the intercept "b" become -1?
please help
to find b, we cancel out mx with y - mx = mx - mx + b, then we get b = y - mx, finally we plugging (x, y) = (1, 1) m=2, b = 1 - 2(1) we get -1
Disculpe profresor me encanta sus videos un favor sera que puede poner subtitulos en español porfavor
Puedes habilitar los subtítulos automáticos en español. No son perfectos, pero deberían ser lo suficientemente buenos.
Debajo del video, habilita los subtítulos haciendo clic en el ícono "CC". Luego haga clic en el ícono del engranaje, haga clic en "Subtítulos / CC", seleccione "Traducción automática" y luego seleccione "Español".
@@domenicodefelice2083 ¡Excelente respuesta, Domenico! 👋 Seguí tus claras instrucciones pero en mi caso la traducción del ruso al inglés de un video corto en RUclips es *_malísima._* 😂
Sir I want to ask you one question- "what dx in integration notation mean"
pretty sure it just refers to an incredibly small change, dx being change in x,
I dont get why y'=2x-1 would be the derivative of y=x^2, becaus if you apply the rules of diferentiation to y then the result woulde be 2x, can someone explain?
You're correct:
f'x: x^2 ==> fx: 2x
and we have know that (y-intercept) at x = 1, m = 2, Equal to 1 (as showing in video @ 4.13min)
y-intercept = mx + b
y-intercept = 1
m = 2
x = 1
b = ?? (unknown)
So:
1 = (2)(1) + b
doing basic algebra:
b = -2 + 1 = -1
@@unnamed-w4k why di you change 2 into -2 at the bottom
because y = 2x-1 isn't the derivative it's the equation of the tangent
Does a line perpendicular to the curve that goes through a certain point also only share just this one point with the curve? Dont know -probably not ;)
so the word perpendicular can only apply to two lines, not to a curve!
Professor Dave Explains okay thanks! Beyond my scope of understanding momentarily but thanks!
so perpendicular means two lines have a right angle between them, but a curve is changing direction constantly, so we can't define an angle between a curve and a line
Professor Dave Explains new day, new brain.i get it. If you'd represent a curve just by aligned dots one would notice that if a line that had the exact thickness of the dot would always cut one of the other dots because theyre kind of laterally in front of the middle dot. But what about a curve that changes direction? Haha! Thanks!
whoa, sorry i don't understand at all! all curves change direction, if they didn't, they'd be lines.
Te rifaste krnal
Secant line is a straight line, it must have a constant rate of change or slope. Isn't it?
yea if it didn't it would be a curve
how many times does he said =
ok boss but why do u want to find the slope of a tangent line?
to find the rate at which something is changing.
thanks boss
my professor skipped this part
😀😀😀😀😀
But the derivative of y=x*2. Is 2x not 2x-1
Please anyone
Do you mean the derivative of x²? In that case, it would be 2x by the power rule:
power rule states the derivative of x^n to be n(x^(n-1)) where n is the exponent of x:
d(x^n)/dx = n(x^(n-1))
Therefore,
d(x²)/dx = 2(x^(2 - 1)) = 2(x^1)
x^1 is just x:
d(x²)/dx = 2x
The case of 2x - 1 is not the derivative or slope of the line. It is in the linear equation of:
y = 2x - 1
This equation is simply the tangent line's slope-intercept form. The slope of this line will be the derivative of the point of intersection of y = 2x - 1 and y = x² ( (1, 1) in this case):
d(x²)/dx with x as 1 = tangent line's slope = 2
this is proven with d(x²)/dx = 2x (the equation with the power rule):
plug in x = 1
d(1²)/dx = 2(1) = 2 // same answer
Woah
Our good old friend y=x^2😅
KFUPM in Saudi
This isn't as complex as I thought
It's hilarious because i don't speak English so well and I learn better than spanish videos