### Video Transcript

Find d by dπ₯ of the integral evaluated between five and π₯ to the fourth power of the natural log of seven plus π‘ cubed with respect to π‘.

Now, as tempting as it may be, we absolutely do not want to evaluate the integral here. Instead, weβ²re going to recall the second fundamental theorem of calculus. This says that if π is a continuous function and π is a constant, then the integral evaluated between π and π₯ of π of π‘ with respect to π‘ is equal to capital πΉ of π₯ minus capital πΉ of π, where capital πΉ prime of π₯ is equal to the function of π₯. And remember, capital πΉ is the antiderivative of the function. So how does this help us?

Weβ²re going to differentiate both sides of the equation with respect to π₯. And we see that we can separate our derivative into d by dπ₯ of capital πΉ of π₯ minus d by dπ₯ of capital πΉ of π. Remember, capital πΉ of π will give us a constant. This means itβ²s independent of π₯. And so, its derivative is going to be zero. So weβre left with the derivative of capital πΉ of π₯ with respect to π₯ or capital πΉ prime of π₯. Earlier though, we said that capital πΉ prime of π₯ was equal to π of π₯. And we therefore see that the derivative with respect to π₯ of the integral evaluated between π and π₯ of π of π‘ with respect to π‘ is equal to π of π₯.

Letβ²s see how we can use this to evaluate the derivative with respect to π₯ of the integral between five and π₯ to the fourth power of the natural log of seven plus π‘ cubed with respect to π‘. We can say that in our example, π of π‘ is equal to the natural log of seven plus π‘ cubed. And we use the rules we developed earlier. And we see that the derivative with respect to π₯ is equal to d by dπ₯ of capital πΉ of π₯ to the fourth power minus capital πΉ of five. We split our derivative up. And then we recall that the derivative of capital πΉ of five with respect to π₯ is going to be equal to zero, since itβ²s going to be independent of π₯.

And so, we see that the derivative with respect to π₯ of the integral evaluated between five and π₯ to the fourth power of the natural log of seven plus π‘ cubed with respect to π‘ is the derivative of capital πΉ of π₯ to the fourth power with respect to π₯. Weβ²re going to need to be really careful here when evaluating d by dπ₯ of capital πΉ of π₯ to the power of four. Weβ²re going to be differentiating a function of a function, or a composite function. So weβ²re going to need to use the chain rule.

This says that if π¦ is a function in π’ and π’ itself is a function in π₯, then the derivative of π¦ with respect to π₯ is equal to dπ¦ by dπ’ times dπ’ by dπ₯. So this means that the derivative of capital πΉ of π₯ to the power of four with respect to π₯ is equal to the derivative of capital πΉ of π₯ to the power of four, which is just πΉ prime of π₯ to the power of four multiplied by the derivative of π₯ to the power of four with respect to π₯. Thatβs four π₯ cubed.

Now, we said earlier that πΉ prime of π₯ was equal to π of π₯. So we can say that πΉ prime of π₯ to the power of four is equal to π of π₯ to the power of four. Since we said that π of π‘ was equal to the natural log of seven plus π‘ cubed, we know that π of π₯ to the fourth power is equal to the natural log of seven plus π₯ to the fourth power cubed times four π₯ cubed. π₯ to the fourth power cubed is π₯ to the power of 12. And we can see that the derivative with respect to π₯ of the integral evaluated between five and π₯ to the fourth power of the natural log of seven plus π‘ cubed with respect to π‘ is equal to four π₯ cubed times the natural log of seven plus π₯ to the power of 12.