 Welcome back everyone to lecture 46 in our series math 12-10. We've seen in previous videos, well we've learned about this fundamental theorem of calculus part one, which tells us that the derivative of an integral is equal to the original function, the integrand inside of there. More specifically, if we take the derivative with respect to x of the function, some constant a to the x, the integral from a to x of f of t dt, that this is equal to just f of x. And so what I want to do in this video here is look at how you can use FTC1 to help us calculate some derivatives of integral functions, which might not match up perfectly with the format that FTC1 requires. So if you look at this one right here, let's find the derivative with respect to x of the integral x to 1 t squared here dt. Now the thing I want to point out to you here is that with this integral, the limits are in the wrong locations right here, right? With FTC1, you want a constant on the bottom and you need a variable on the top as you can see there. But with this set in, we have the variable on the bottom and the constant on the top. In order to apply FTC1, we need to switch the order of these things. We would need to switch these things around. And it turns out that's actually a very simple trick to do with integrals here because if ever you have an integral like a to b of f of x dx, you can always switch the order by negating the integral. So by taking negative the integral from b to a of f of x dx, that'll be equal to what we were having before. So if you apply that to the original integral, we want to take the derivative with respect to x. But now we're going to get negative the integral from a, I guess it's a specific number from one to x, t squared dt. And as we're taking the derivative, and this is a constant multiple, you can take that negative sign out. So you integrate from one to x, t squared dt, take the derivative. So then by FTC1, the fundamental theorem calculus part one, if you take the derivative of an integral where the variable x is on the top and the constant is on the bottom, by that rule, you're going to get negative x squared. And so that's the thing to remember here is you need that negative sign, otherwise your derivative is going to be off by a little bit. And so make sure that you have a variable in the upper bound of your integral, otherwise your derivative calculation will be incorrect. There are some other subtleties that come up when one looks at taking derivatives using FTC1. And we will see some of those in the next video. See you then.