Given {eq}f(x) = \int _x ^{x^2} t ^2 dt {/eq}.

(a) Find f'(x).

(b) Find f'(2).

Question:

Given {eq}f(x) = \int _x ^{x^2} t ^2 dt {/eq}.

(a) Find f'(x).

(b) Find f'(2).

Leibniz Rule:

Consider some function {eq}\displaystyle g {/eq} such that it is continuous on the interval {eq}\displaystyle [a,b] {/eq} and let {eq}\displaystyle p(x) {/eq} and {eq}\displaystyle q(x) {/eq} be functions which are differentiable with respect to {eq}\displaystyle x {/eq} for all {eq}\displaystyle x\in[a,b] {/eq}. Then according to the Leibniz rule we will have

$$\displaystyle \frac{d}{dx}\int_{p(x)}^{q(x)}g(t) \ dt=g(q(x))\frac{d}{dx}[q(x)]-g(p(x))\frac{d}{dx}[p(x)] $$

If in the above formula we have {eq}\displaystyle p(x)=c {/eq} (where, {eq}\displaystyle c {/eq} is some constant) and {eq}\displaystyle q(x)=x {/eq}, then the above equation reduces to the fundamental theorem of calculus.

Answer and Explanation: 1

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(a)

Using the Leibniz rule, we can evaluate the derivative as shown below

{eq}\displaystyle \begin{align} f'(x)&=\frac{d}{dx}\left[...

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The Fundamental Theorem of Calculus

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


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