Chapter 2: Q. 52 (page 222)

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)}$

Short Answer

Expert verified

The derivative of function is $y' = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)} [2 + \frac{12 x^{2}}{x^{3} - 2} - \frac{1}{x} - \frac{5 e^{5 x}}{3 e^{5 x} + 1}]$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)}$

Step 2. Calculation

First, we will take the log on both sides and then differentiate it,

$\ln y = \ln [e^{2 x} {(x^{3} - 2)}^{4}] - \ln (x (3 e^{5 x} + 1)) \ln y = \ln (e^{2 x}) + \ln {(x^{3} - 2)}^{4} - \ln x - \ln (3 e^{5 x} + 1) \ln y = 2 x + 4 \ln (x^{3} - 2) - \ln x - \ln (3 e^{5 x} + 1) \frac{1}{y} y' = 2 + \frac{4}{x^{3} - 2} (3 x^{2}) - \frac{1}{x} - \frac{1}{3 e^{5 x} + 1} (5 e^{5 x}) y' = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)} [2 + \frac{12 x^{2}}{x^{3} - 2} - \frac{1}{x} - \frac{5 e^{5 x}}{3 e^{5 x} + 1}]$

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Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

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Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58. f(x)=e2xx3-24x3e5x+1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \frac{e^{2 x} {(x^{3} - 2)}^{4}}{x (3 e^{5 x} + 1)}$