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# Chain rule introduction

An introductory example of the chain rule for the function (sin(x))^2. Created by Sal Khan.

Video transcript

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 of anything even reasonably complex. And it's called the chain rule. And when you're
first exposed to it, it can seem a little daunting
and a little bit convoluted. But as you see more
and more examples, it will start to make sense. And hopefully it
even starts to seem a little bit simple and
intuitive over time. So let's say that
I have a function. 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 squared. Now I could have written that
like this, sine squared of x. But it will be a
little bit clearer using that type of notation. So let me make it
so I have h of x. And what I'm curious about
is, what is h prime of x? So I want to know h prime of x,
which a another way of writing it is, the derivative
of h with respect to x. These are just
different notations. And to do this, I'm going
to use the chain rule. And the chain rule
comes into play any time your
function can be used as a composition of
more than one function. And that might not
seem obvious right now, but it will hopefully
maybe by the end of this video or the next one. Now what I want to
do is a little bit of a thought experiment. If I were to ask you,
what is the derivative with respect to
x, if I were just to apply the derivative operator
to x squared with respect to x, what do I get? Well, this gives me 2x. We've seen that many, many,
many, many, many times. Now what if I were take
the derivative with respect to a of a squared? Well it's the exact same thing. I just swapped an a for the x's. This is still going
to be equal to 2a. Now I will do
something that might be a little bit more bizarre. What if I were take the
derivative with respect to sine of x of
sine of x squared? Well, wherever I had the x's
up here, or the a's over here, I just replace it
with a sine of x. So this is just going to be
2 times the thing that I had. So whatever I'm taking the
derivative with respect to. Here it was with respect to x. Here it was with respect to a. Here it's with
respect to sine of x. So it's going to be
2 times sine of x. Now, so the chain rule tells us
that this derivative is going to be the derivative of our
whole function with respect-- or the derivative of
this outer function, x squared-- the derivative
of this outer function with respect to sine of x. So that's going
to be 2 sine of x. So we could view it
as the derivative of the outer
function with respect to the inner, 2 sine of x. We could just treat sine of
x like it's kind of an x, and it would have been just 2x. But instead it's a sine of
x, and we say 2 sine of x. Times the derivative-- let
me do this in green-- times the derivative of sine
of x with respect to x. Well, that's more
straightforward and a little bit more intuitive. The derivative of
sine of x with respect to x, we've seen multiple
times, is cosine of x. So times cosine of x. And so there we've
applied the chain rule. It was the derivative of the
outer function with respect to the inner. So the derivative of sine
of x squared with respect to sine of x is 2x sine of x. And then we multiply
that times the derivative of sine of x with respect to x. So let me make it clear. This right over here
is the derivative, we're taking the derivative
of sine of x squared. So let me make it
clear, that's what we're taking the derivative of. With respect to sine of x. And then we're multiplying that
times the derivative of sine of x with respect to x. And this is where it
might start making a little bit of intuition. You can't really treat these
differentials, this d whatever, this dx, this d sine
of x, as a number. And you really can't,
because the notation makes it look like a
fraction, because intuitively, that's what we're doing. But if you were to treat
then like fractions, then you could think about
canceling that and that. And once again, this isn't
a rigorous thing to do, but it can help
with the intuition. And then what
you're left with is the derivative of
this whole sine of x squared with respect to x. So you're left
with the derivative of, essentially our
original function, sine of x squared with respect to x. Which is exactly what dh/dx is. This right over here is
our original function, h. That's our original function h. So it might seem a
little bit daunting now. What I'll do in the next video
is another several examples. And then we'll try to
abstract this a little bit.