Applying the chain rule twice | Advanced derivatives | AP Calculus AB | Khan Academy

Let's say that y is equal to sine of x squared to the third power, which of course we could also write as sine of x squared to the third power. What we're curious about is what is the derivative of this with respect to x? What is dy/dx, which we could also write as y prime?

Well, there's a couple of ways to think about it. This isn't a straightforward expression here, but you might notice that I have something being raised to the third power. In fact, if we look at the outside of this expression, we have some business in here and it's being raised to the third power.

One way to tackle this is to apply the chain rule. So, if we apply the chain rule, it's going to be the derivative of the outside with respect to the inside, or the something to the third power. The derivative of the something to the third power with respect to that something is going to be 3 times that something squared times the derivative with respect to x of that something. In this case, the something is sine.

Let me write that in blue color. It is sine of x squared. It is sine of x squared! No matter what was inside of these orange parentheses, I would put it inside of the orange parentheses and these orange brackets right over here. We learned that in the chain rule, so let's see.

We know this is just a matter of algebraic simplification, but the second part we need to now take the derivative of sine of x squared. Well, now we would want to use the chain rule again. So, I'm going to take the derivative. It’s sine of something, so this is going to be the derivative of this is going to be the sine of something with respect to something.

That is cosine of that something times the derivative with respect to x of the something. In this case, the something is x squared. And of course, we have all of this out front, which is the 3 times sine of x squared, and I could write it like this squared.

All right, so we're getting close. Now we just have to figure out the derivative with respect to x of x squared. We've seen that many times before; we just use the power rule. That's going to be 2x.

So if we wanted to write the dy/dx, we get a little bit of a mini drum roll here. This didn't take us too long! dy/dx—I'll multiply the 3 times the 2x, which is going to be 6x.

So I covered those so far times sine squared of x squared times cosine of x squared, and we are done with applying the chain rule multiple times!