Chapter 2: Q. 49 (page 222)

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \sqrt{x \ln | 2^{x} + 1 |}$

Short Answer

Expert verified

The derivative of function is $y' = \frac{1}{2} \sqrt{x \ln | 2^{x} + 1 |} [\frac{1}{x} + \frac{1}{\ln (2^{x} + 1)} \cdot \frac{1}{2^{x} + 1} \ln 2 \cdot 2^{x}]$

Step by step solution

Step 1. Given Information

The given function is $f (x) = \sqrt{x \ln | 2^{x} + 1 |}$

Step 2. Calculation

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

$\ln y = \ln \sqrt{x \ln | 2^{x} + 1 |} \ln y = \frac{1}{2} [\ln (x \ln | 2^{x} + 1 |)] \ln y = \frac{1}{2} [\ln x + lnln | 2^{x} + 1 |] \frac{1}{y} y' = \frac{1}{2} [\frac{1}{x} + \frac{1}{\ln (2^{x} + 1)} (\frac{1}{2^{x} + 1}) \ln 2 (2^{x})] y' = \frac{1}{2} \sqrt{x \ln | 2^{x} + 1 |} [\frac{1}{x} + \frac{1}{\ln (2^{x} + 1)} (\frac{1}{2^{x} + 1}) \ln 2 (2^{x})]$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \sqrt{x \ln | 2^{x} + 1 |}$

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

Calculus

Decision Maths

Logic and Functions

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.f(x)=xln|2x+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

Calculus

Decision Maths

Logic and Functions

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.
$f (x) = \sqrt{x \ln | 2^{x} + 1 |}$