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Chapter 2: Q. 68 (page 234)

We have to find the function of given derivative
$f^{'} (x) = \frac{1}{1 + 9 x^{2}}$

Short Answer

Expert verified

The given function is the answer of given derivative

f (x) = \frac{1}{3} \tan^{- 1} 3 x

Step by step solution

01

Step 1.Given information

We have been given the derivative function

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

02

Step 2.Finding the approximate function of derivative

One can think of $\tan^{- 1} x$ as the denominator is suggesting about this function

\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}

03

Step 3.Finding more appropriate function

$f (x) = \frac{1}{3} \tan^{- 1} (3 x^{})$

Here we have used product rule

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

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Most popular questions from this chapter

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

$a) f (x) = {(3 x + 1)}^{4} b) f (x) = {(\frac{1}{x} + \frac{1}{x^{2}})}^{2} c) f (x) = {(x + 1)}^{2} \sqrt{x} d) f (x) = \frac{{(x + 1)}^{2}}{\sqrt{x}}$

Use the definition of the derivative to find $f'$ for each function $f$ in Exercises 34-59

role="math" localid="1648284617718" $f (x) = \sqrt{2 x + 1}$

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?

A tomato plant given $x$ ounces of fertilizer will successfully bear $T (x)$ pounds of tomatoes in a growing season.

(a) In real-world terms, what does $T (5)$ represent and what are its units? What does $T' (5)$ represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than $20$ ounces are used, but inhibits production when more than $20$ ounces are used. When is $T (x)$ positive and when is $T (x)$ negative? When is $T' (x)$ positive and when is $T' (x)$ negative?

Find the derivative of the absolute value function and piecewise defined function

$f (x) = \{\begin{cases} 3 x + 1 & i f x \leq 1 \\ x^{3} & i f x > 1 \end{cases}$

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