Q. 69 Find a function f from its deriv... [FREE SOLUTION]

Chapter 2: Q. 69 (page 234)

Find a function f from its derivative
$f^{鈥�} (x) = \frac{3 x}{1 + 9 x^{2}}$

Short Answer

Expert verified

The answer of given derivative function is

f (x) = \frac{1}{6} \ln 鈦� (1 + 9 x^{2})

Step by step solution

Step 1.Given information

We have been given the derivative of function

f^{鈥�} (x) = \frac{3 x}{1 + 9 x^{2}}

Step 2. Finding the function

One can suspect from the derivative that function $\tan^{鈭� 1} 鈦� x$

\begin{aligned} f (x) = \tan^{鈭� 1} 鈦� 3 x \\ f (x) = \tan^{鈭� 1} 鈦� 3 x \\ f^{鈥�} (x) = \frac{1}{1 + (3 x)^{2}} \frac{d}{d x} (3 x) \\ = \frac{3}{1 + 9 x^{2}} \end{aligned}

Step 3. Getting more approximate function

Now the approximate function is $f (x) = \frac{1}{6} \ln 鈦� (1 + 9 x^{2})$

\begin{aligned} f (x) = \frac{1}{6} \ln 鈦� (1 + 9 x^{2}) \\ f^{鈥�} (x) = \frac{1}{6} \frac{d}{d x} \ln 鈦� (1 + 9 x^{2}) \\ = \frac{1}{6 (1 + 9 x^{2})} \frac{d}{d x} (1 + 9 x^{2}) [\frac{d}{d x} \ln 鈦� x = \frac{1}{x}] \\ = \frac{1}{6 (1 + 9 x^{2})} (\frac{d}{d x} 1 + \frac{d}{d x} 9 x^{2}) \\ = \frac{9 (2 x)}{6 (1 + 9 x^{2})} \\ = \frac{3 x}{(1 + 9 x^{2})} \end{aligned}

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

Find a function f from its derivative
$f^{鈥�} (x) = \frac{3 x}{1 + 9 x^{2}}$

Short Answer

Step by step solution

Step 1.Given information

Step 2. Finding the function

Step 3. Getting more approximate function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Find a function f from its derivativef鈥�(x)=3x1+9x2

Short Answer

Step by step solution

Step 1.Given information

Step 2. Finding the function

Step 3. Getting more approximate function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find a function f from its derivative
$f^{鈥�} (x) = \frac{3 x}{1 + 9 x^{2}}$