### Video Transcript

Calculate the derivative with respect to π₯ of the integral from zero to π₯ cubed of cos to the power of four of π‘ with respect to π‘.

Weβre being asked to work out the derivative of an integral with a function of π₯ as one of its limits. But this is not as complicated as it looks. In fact, we can use a generalization of the fundamental theorem of calculus. This says that if π’ is a differentiable function of π₯ and π is continuous, then the derivative with respect to π₯ of the integral between π and π’ of π of π‘ with respect to π‘ is equal to π of π’ times dπ’ by dπ₯. We can apply this directly to our problem since, in our case, π’ is equal to π₯ cubed, π is equal to zero, and π of π‘ is equal to cos to the power of four of π‘.

In order to work out the derivative of the given integral, we therefore need to find only π of π’ and dπ’ by dπ₯. Now, with π’ equal to π₯ cubed, dπ’ by dπ₯, by the power rule, is equal to three π₯ squared. And with π of π‘ equal cos to the power four of π‘, since π‘ is a dummy variable, we have π of π’ equals cos to the power of four of π’. And since π’ is equal to π₯ cubed, this is equal to cos to the power of four of π₯ cubed.

So now, we have the derivative with respect to π₯ of the integral between zero and π₯ cubed of cos to the power of four of π‘ dπ‘ is equal to π of π’ dπ’ by dπ₯. Where π of π’ is equal to cos to the power of four of π₯ cubed, and dπ’ by dπ₯ is three π₯ squared. If we rearrange this, we have three π₯ squared times cos to the power of four of π₯ cubed.