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Find the derivative of the function F ( x ) = ?^e^x_x 1/ t d t .

Question:

Find the derivative of the function

{eq}F(x)=\int_{x}^{e^{x}}\frac{1}{t}dt. {/eq}

Fundamental Theorem of Calculus

Integration and differentiation are related in a crucial way, and the Fundamental Theorem of Calculus explains this relationship. If we want to take the derivative of an indefinite integral, we can thus apply this Theorem. Depending on the bounds of the integral, we may also need to use the Chain Rule.

{eq}\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) dt = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x) {/eq}

Answer and Explanation:

The function given to us is defined by an integral. While we could certainly integrate to find this function and then differentiate the result, it's actually quicker to apply the Fundamental Theorem of Calculus. Let's begin by finding a variety of necessary functions and derivatives based on the integrand and bounds of integration.

{eq}\begin{align*} &f(t) = \frac{1}{t}\\ &g(x) = x && f(g(x)) = \frac{1}{x} && g'(x) = 1\\ &h(x) = e^x && f(h(x)) = \frac{1}{e^x} = e^{-x} && h'(x) = e^x \end{align*} {/eq}


We can now combine the information to find the derivative of this function.

{eq}\begin{align*} F'(x) &= e^{-x}e^x - \frac{1}{x}\\ &= 1 - \frac{1}{x} \end{align*} {/eq}


Learn more about this topic:

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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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