Name: Chain rule | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-08-17T06:40:05.000Z
Duration: 5 min 7 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

- [Instructor] What we're going to go over

in this video is one of the core principles in calculus,

and you're going to use it any time you take the derivative,

it can seem a little daunting and a little bit convoluted.

it'll start to make sense, and hopefully it'd even start

to seem a little bit simple and intuitive over time.

Let's say I have a function h of x, and it is equal to,

just for example, let's say it's equal to sine of x,

let's say it's equal to sine of x squared.

sine squared of x, but it'll be a little bit clearer

And what I'm curious about is what is h prime of x?

And to do this, I'm going to use the chain rule.

and the chain rule comes into play every time,

any time your function can be used as a composition

And as that might not seem obvious right now,

maybe by the end of this video or the next one.

what is the derivative with respect to x,

if I were to just apply the derivative operator

to x squared with respect to x, what do I get?

We've seen that many, many, many, many times.

Now, what if I were to take the derivative with respect to a

This is still going to be equal to two a.

What if I were to take the derivative with respect to

Well, wherever I had the x's up here, the a's over here,

So this is just going to be two times the thing that I had,

so whatever I'm taking the derivative with respect to.

that this derivative is going to be the derivative

or the derivative of this outer function, x squared,

So we could view it as the derivative of the outer function

with respect to the inner, two sine of x.

We could just treat sine of x like it's kind of an x.

times the derivative of sine of x with respect to x.

Times the derivative of sine of x with respect to x,

with respect to x, we've seen multiple times,

And so there we've applied the chain rule.

It was the derivative of the outer function

with respect to sine of x is two sine of x,

and then we multiply that times the derivative of sine of x

we're taking the derivative of sine of x squared.

That's what we were taking the derivative of

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Chain rule | Derivative rules | AP Calculus AB | Khan Academy

essentially

obvious

multiple

intuitive