Finding derivative with fundamental theorem of calculus | AP®︎ Calculus AB | Khan Academy

Let's say that we have the function g of x, and it is equal to the definite integral from 19 to x of the cube root of t dt. What I'm curious about finding, or trying to figure out, is what is g prime of 27? What is that equal to? Pause this video and try to think about it, and I'll give you a little bit of a hint. Think about the second fundamental theorem of calculus.

All right, now let's work on this together. So we want to figure out what g prime is. We could try to figure out what g prime of x is and then evaluate that at 27. The best way that I can think about doing that is by taking the derivative of both sides of this equation.

So, let's take the derivative of both sides of that equation. The left-hand side, we'll take the derivative with respect to x of g of x, and the right-hand side, the derivative with respect to x of all of this business. Now, the left-hand side is pretty straightforward. The derivative with respect to x of g of x, that's just going to be g prime of x.

But what is the right-hand side going to be equal to? Well, that's where the second fundamental theorem of calculus is useful. I'll write it right over here: second fundamental theorem of calculus. It tells us, let's say we have some function capital F of x, and it's equal to the definite integral from a, some constant a, to x, of lowercase f of t dt.

The second fundamental theorem of calculus tells us that if our lowercase f is continuous on the interval from a to x, so I'll write it this way on the closed interval from a to x, then the derivative of our capital F of x, so capital F prime of x, is just going to be equal to our inner function f, evaluated at x instead of t. It's going to become lowercase f of x.

Now, I know when you first saw this, you thought that, hey, this might be some cryptic thing that you might not use too often. But we're going to see that it's actually very, very useful. And even in the future, some of you might already know there's multiple ways to try to think about a definite integral like this, and you'll learn it in the future.

But this can be extremely simplifying, especially if you have a hairy definite integral like this. And so this just tells us, hey, look, the derivative with respect to x of all of this business. First, we have to check that our inner function, which would be analogous to our lowercase f here, is continuous on the interval from 19 to x.

Well, no matter what x is, this is going to be continuous over that interval because this is continuous for all x's. And so we meet this first condition, our major condition. And so then we could just say, all right, then the derivative of all of this is just going to be this inner function, replacing t with x. So we're going to get the cube root. Instead of the cube root of t, you're going to get the cube root of x.

And so we can go back to our original question: what is g prime of 27 going to be equal to? What's going to be equal to the cube root of 27, which is of course equal to 3, and we're done.