Find {eq}f'(x) {/eq} if {eq}\displaystyle f(x) = \int_0^{4x} \frac {dt}{t^5 + 1} {/eq}.

## Question:

Find {eq}f'(x) {/eq} if {eq}\displaystyle f(x) = \int_0^{4x} \frac {dt}{t^5 + 1} {/eq}.

## Fundamental Theorems of Calculus

There are two fundamental theorems of calculus. The first fundamental theorem states that:

{eq}\int_{a}^{b} f(x) dx = F(b)-F(a) {/eq}

where F(x) is the antiderivative of f(x). The second fundamental theorem states that given:

{eq}F(x)=\int_{a}^{x} f(t) dt {/eq}

we have the derivative given by {eq}F'(x)=f(x) {/eq}. As with the rules for differentiation, remembering these two theorems involving integrals can greatly simplify integration questions.

## Answer and Explanation: 1

Given {eq}f(x)=\int_{0}^{4x} \frac{1}{t^5+1} dt {/eq}, we can apply the second fundamental theorem of calculus to get:

{eq}f'(x)=\frac{1}{(4x)^5+1} {/eq}

This can be simplified to:

{eq}f'(x)=\frac{1}{1024x^5+1} {/eq}

#### Learn more about this topic:

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Chapter 12 / Lesson 10Learn about the fundamental theorem of calculus. Identify how to prove the fundamental theorem, and examine graphs and examples of several ways to use it.