Chapter 2: Q. 72 (page 234)

In Exercises 72–77, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process. (Some of these exercises involve hyperbolic functions.)
$f' (x) = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$

Short Answer

Expert verified

The function that has the derivative $f' (x) = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$ is $f (x) = \frac{1}{2} \sqrt{1 + 4 x^{2}}$ .

Step by step solution

Step 1. Given Information.

The derivative:

$f' (x) = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$

Step 2. Guess the formula for the given derivative.

We know that,

$\frac{d}{d x} (\sin h^{- 1} x) = \frac{1}{\sqrt{1 + x^{2}}}$

So we can conclude that the derivative will be of the form of hyperbolic sinefunction.

Step 3. Check guess-and-check process.

$f (x) = \sin h^{- 1} (2 x) f' (x) = \frac{1}{\sqrt{1 + x^{2}}} . (2) = \frac{2}{\sqrt{1 + x^{2}}}$

which is not a given derivative.

Step 4. Check another function.

Consider the function,

$f (x) = \frac{1}{2} \sqrt{1 + 4 x^{2}} f' (x) = \frac{d}{d x} (\frac{1}{2} {(1 + 4 x^{2}))}^{\frac{1}{2}} f' (x) = \frac{1}{2} . [\frac{1}{2} {(1 + 4 x^{2})}^{- \frac{1}{2}} . 8 x] = \frac{1}{2} [\frac{4 x}{\sqrt{1 + 4 x^{2}}}] = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$

which is the given derivative.

So the function is $f (x) = \frac{1}{2} \sqrt{1 + 4 x^{2}}$

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In Exercises 72–77, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process. (Some of these exercises involve hyperbolic functions.)
$f' (x) = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Guess the formula for the given derivative.

Step 3. Check guess-and-check process.

Step 4. Check another function.

One App. One Place for Learning.

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In Exercises 72–77, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process. (Some of these exercises involve hyperbolic functions.)f'(x)=2x1+4x2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Guess the formula for the given derivative.

Step 3. Check guess-and-check process.

Step 4. Check another function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Calculus

Mechanics Maths

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 72–77, find a function f that has the given derivative f. In each case you can find the answer with an educated guess-and-check process. (Some of these exercises involve hyperbolic functions.)
$f' (x) = \frac{2 x}{\sqrt{1 + 4 x^{2}}}$