Applying the chain rule on an integral?

Question

I am currently practicing taking the derivatives of functions and I am familiar with the rules, but when it comes to integrals I am stuck.

For example:

$$g(x) = \int_{1}^{x^4} \sec{t} \, dt $$

In the book they say to use the chain rule, but what exactly are we applying the chain rule to? Are we applying the chain rule to $$\sec(t)$$ or are we applying it to the upper limit $$x^4 ?$$

I am just confused because the book substitutes the upper limit with the letter u which leads me to believe they are taking the chain rule of the upper limit, but why do that?

Answer 1

Hint: Think of $g$ as the composition $f \circ h$, where $$h(x) := x^4 \qquad \text{and} \qquad f(u) := \int_1^u \sec t \,dt,$$ and apply the usual Chain Rule directly (computing $f'(u)$ is just a matter of using the statement of the F.T.C.).