Q. 50 Use the definition of the deriva... [FREE SOLUTION]

91影视

Chapter 10: Q. 50 (page 777)

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54
$f (x) = \frac{1}{\sqrt{2 x + 1}}$

Short Answer

Expert verified

The value of $f' (x) = - {(2 x + 1)}^{- \frac{3}{2}}$

Step by step solution

Step 1. Given information

The given function $f (x) = \frac{1}{\sqrt{2 x + 1}}$

Step 2. Finding the value of f'(x)

We know that $f' (x) = \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$

Given $f (x) = \frac{1}{\sqrt{2 x + 1}}$

Then role="math" localid="1648288671514" $f (x + h) = \frac{1}{\sqrt{2 a + 2 h + 1}}$

$f' (x) = \lim_{h 鈫� 0} \frac{\frac{1}{\sqrt{2 x + 2 h + 1}} - \frac{1}{\sqrt{2 x + 1}}}{h} = \lim_{h 鈫� 0} \frac{\sqrt{2 x + 1} - \sqrt{2 x + 2 h + 1}}{h \sqrt{2 x + 2 h + 1} \sqrt{2 x + 1}} = \lim_{h 鈫� 0} \frac{(\sqrt{2 x + 1} - \sqrt{2 x + 2 h + 1}) (\sqrt{2 x + 1} + \sqrt{2 x + 2 h + 1)}}{h \sqrt{2 x + 2 h + 1} \sqrt{2 a + 1} (\sqrt{2 x + 1} + \sqrt{2 x + 2 h + 1})} = \lim_{h 鈫� 0} \frac{(2 x + 1) - (2 x + 2 h + 1)}{h \sqrt{2 x + 2 h + 1} \sqrt{2 x + 1} (\sqrt{2 x + 1} + \sqrt{2 x + 2 h + 1})} = \lim_{h 鈫� 0} \frac{- 2 h}{h \sqrt{2 x + 2 h + 1} \sqrt{2 x + 1} (\sqrt{2 x + 1} + \sqrt{2 x + 2 h + 1})} P u t t i n g h = 0 = \frac{- 2}{\sqrt{2 x + 1} \sqrt{2 x + 1} (\sqrt{2 x + 1} + \sqrt{2 x + 1})} = \frac{- 1}{{(2 x + 1)}^{\frac{3}{2}}} = - {(2 x + 1)}^{- \frac{3}{2}}$

Hence, the value of $f' (x) = - {(2 x + 1)}^{- \frac{3}{2}}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54
$f (x) = \frac{1}{\sqrt{2 x + 1}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of f'(x)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

91影视

Use the definition of the derivative to find f'for each function fin Exercises 39-54f(x)=12x+1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of f'(x)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 39-54
$f (x) = \frac{1}{\sqrt{2 x + 1}}$