Chapter 2: Q. 60 (page 234)

Find the derivatives of each of the functions in Exercises 51–62. In some cases it may be convenient to do some preliminary algebra. (These exercises involve hyperbolic functions and their inverses.)
$f (x) = x \sqrt{\tanh^{- 1} x}$

Short Answer

Expert verified

The derivative is $\sqrt{\tanh^{- 1} x} + \frac{x}{2 (1 - x^{2}) \sqrt{\tanh^{- 1} x}} .$

Step by step solution

Step 1. Given information.

The given function is,

$f (x) = x \sqrt{\tanh^{- 1} x}$

Step 2. Preliminary knowledge.

We know,

$\frac{d}{d x} (\tanh^{- 1} x) = \frac{1}{1 - x^{2}} \frac{d}{d x} x^{n} = n x^{n - 1}$

Step 3. Derivative.

The derivative of the function is:

$\frac{d}{d x} x \sqrt{\tanh^{- 1} x} = (\frac{d}{d x} x) \sqrt{\tanh^{- 1} x} + x (\frac{d}{d x} \sqrt{\tanh^{- 1} x}) = \sqrt{\tanh^{- 1} x} + x (\frac{d}{d x} {(\tanh^{- 1} x)}^{\frac{1}{2}}) = \sqrt{\tanh^{- 1} x} + x (\frac{1}{2} {(\tanh^{- 1} x)}^{- \frac{1}{2}} \frac{d}{d x} (\tanh^{- 1} x)) [\frac{d}{d x} x^{n} = n x^{n - 1}] = \sqrt{\tanh^{- 1} x} + x (\frac{1}{2 \sqrt{\tanh^{- 1} x}} (\frac{1}{1 - x^{2}})) = \sqrt{\tanh^{- 1} x} + \frac{x}{2 (1 - x^{2}) \sqrt{\tanh^{- 1} x}}$

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Find the derivatives of each of the functions in Exercises 51–62. In some cases it may be convenient to do some preliminary algebra. (These exercises involve hyperbolic functions and their inverses.)
$f (x) = x \sqrt{\tanh^{- 1} x}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Preliminary knowledge.

Step 3. Derivative.

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Find the derivatives of each of the functions in Exercises 51–62. In some cases it may be convenient to do some preliminary algebra. (These exercises involve hyperbolic functions and their inverses.)f(x)=xtanh-1x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Preliminary knowledge.

Step 3. Derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the functions in Exercises 51–62. In some cases it may be convenient to do some preliminary algebra. (These exercises involve hyperbolic functions and their inverses.)
$f (x) = x \sqrt{\tanh^{- 1} x}$