 In this video we provide the solution to question number three for the practice final exam for Math 12-10. We're asked to compute the derivative of the integral from 0 to x to the fourth times cosine squared theta d theta. So to calculate the derivative here we're going to have to use the fundamental theorem of calculus part one, which tells us that the derivative of an integral is the original function. But since the upper limit here is not x and we're taking the derivative with respect to x the chain rule is going to come into play. So notice that your inner function is going to be x to the fourth and your outer function is going to be this integral function right here. So we have to calculate the derivative using that chain rule so we can think of it in the following terms. We're going to take the derivative with respect to x to the fourth of the integral from 0 to x to the fourth cosine squared theta d theta. And then we're also going to times that by the derivative of x to the fourth over the with respect to x we should say it that way. So then by FTC1 the derivative of the integral will just give us back cosine squared of x to the fourth. So that inner function goes inside and replaces the theta. Then we have to take the derivative of x to the fourth which will be 4x cubed in which case then we search and see that d would then be the correct answer.