Chapter 2: Q. 69 (page 234)

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

Short Answer

Expert verified

The answer of given derivative function is

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

Step by step solution

Step 1.Given information

We have been given the derivative of function

f^{'} (x) = \frac{3 x}{1 + 9 x^{2}}

Step 2. Finding the function

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

\begin{aligned} f (x) = \tan^{- 1} 3 x \\ f (x) = \tan^{- 1} 3 x \\ f^{'} (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 $f (x) = \frac{1}{6} \ln (1 + 9 x^{2})$

\begin{aligned} f (x) = \frac{1}{6} \ln (1 + 9 x^{2}) \\ f^{'} (x) = \frac{1}{6} \frac{d}{d x} \ln (1 + 9 x^{2}) \\ = \frac{1}{6 (1 + 9 x^{2})} \frac{d}{d x} (1 + 9 x^{2}) [\frac{d}{d x} \ln 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}

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

Short Answer

Step by step solution

Step 1.Given information

Step 2. Finding the function

Step 3. Getting more approximate function

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Find a function f from its derivativef′(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

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Study anywhere. Anytime. Across all devices.

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