Answered: y = (x? + 2)(x* + 4)*

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 31WE

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Question

Use logarithmic differentiation to find the derivative
of the function.

Expert Solution

Step 1

Given Data

The function is $y = {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}$ .

Apply the logarithm on both side fo the function $y = {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}$ and differentiate with respect to x,

$\begin{array}{rcl} \ln y & = & \ln {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4} \\ \frac{d}{d x} (\ln y) & = & \frac{d}{d x} (\ln {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}) \\ \frac{1}{y} \frac{d y}{d x} & = & \frac{1}{{(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}} ({(x^{2} + 2)}^{2} \frac{d}{d x} ({(x^{4} + 4)}^{4}) + {(x^{4} + 4)}^{4} \frac{d}{d x} ({(x^{2} + 2)}^{2})) \\ \frac{d y}{d x} & = & \frac{y}{{(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}} ({(x^{2} + 2)}^{2} (4 {(x^{4} + 4)}^{3} (4 x^{3})) + {(x^{4} + 4)}^{4} (2 (x^{2} + 2) (2 x))) \\ = & \frac{{(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}}{{(x^{2} + 2)}^{2} {(x^{4} + 4)}^{4}} (16 x^{3} {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} + 4 x (x^{2} + 2) {(x^{4} + 4)}^{4}) \\ = & (16 x^{3} {(x^{2} + 2)}^{2} {(x^{4} + 4)}^{3} + 4 x (x^{2} + 2) {(x^{4} + 4)}^{4}) \end{array}$

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