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

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. It's called the chain rule. 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'll start to make sense, and hopefully, it'll even start to seem a little bit simple and intuitive over time.

So, let's say that I had a function; let's say I have a function h of x, and it is equal to, just for example, I let's say it's equal to sine of x. Let's say it's equal to sine of x squared. Now, I could have written that. I could have written it like this: sine squared of x. But it'll 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? I want to know h prime of x, which another way of writing it is the derivative of h with respect to x. These are just different notations. To do this, I'm going to use the chain rule. The chain rule comes into play every time your function can be used as a composition of more than one function. And as that might not seem obvious right now, 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, a little bit of a thought experiment. If I were to ask you what is the derivative, with respect to x, if I were to supply 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 times.

Now, what if I were to 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 to 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 them 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, with respect to x here, with respect to a, here’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 to sine of x. So that's going to be 2 sine of x. 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, so we say 2 sine of x times the derivative—we do this in green—times the derivative of sine of x with respect to x.

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 2 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. 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, with respect to sine of x. And then we're multiplying that times the derivative of sine of x. 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. This notation makes it look like a fraction because intuitively that's what we're doing. But if you were to treat them 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, 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 that a little bit.