Answered: find the first and second derivatives…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.4: Composition Of Functions

Problem 4SE: How do you find the domain for the composition of two functions, fg ?

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Question

find the first and second derivatives of the function

f(x) = (x² + 4)⁵

Expert Solution

Step 1

The given function is $f (x) = {(x^{2} + 4)}^{5}$ .

Step 2

Find the first derivative of the function as shown below.

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} ({(x^{2} + 4)}^{5}) \\ = & 5 {(x^{2} + 4)}^{5 - 1} \cdot \frac{d}{d x} (x^{2} + 4) (By chain rule and \frac{d}{d x} (x^{n}) = n x^{n - 1}) \\ = & 5 {(x^{2} + 4)}^{4} \cdot (2 x) \\ = & 10 x {(x^{2} + 4)}^{4} \end{array}$

Therefore, the first derivative of the function $f (x) = {(x^{2} + 4)}^{5}$ is $\begin{array}{rcl} f' (x) = 10 x {(x^{2} + 4)}^{4} \end{array}$ .

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$Implicit Differentiation$

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