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Chapter 2: Q. 96 (page 236)

Prove each of the differentiation formulas in Exercises 93鈥�96. (These exercises involve hyperbolic functions.)
$\frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{1 - x^{2}}$

Short Answer

Expert verified

We proved the formula $\frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{1 - x^{2}}$ .

Step by step solution

01

Step 1. Given Information

We need to prove the hyperbolic function $\frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{1 - x^{2}}$ .

02

Step 2. Proving the function

$\tan h (\tan h^{- 1} x) = x 鈬� \frac{d}{d x} \tan h (\tan h^{- 1} x) = \frac{d}{d x} (x) 鈬� s e c h^{2} (\tan h^{- 1} x) \frac{d}{d x} (\tan h^{- 1} x) = 1 鈬� \frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{s e c h^{2} (\tan h^{- 1} x)} 鈬� \frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{1 - \tan h^{2} (\tan h^{- 1} x)} 鈬� \frac{d}{d x} (\tan h^{- 1} x) = \frac{1}{1 - x^{2}}$

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Most popular questions from this chapter

Write down a rule for differentiating a composition $f (u (v (w (x))))$ of four functions

(a) in 鈥減rime鈥� notation and

(b) in Leibniz notation.

In the text we noted that if $f (u (v (x)))$ was a composition of three functions, then its derivative is $\frac{d f}{d x} = \frac{d f}{d u} 脳 \frac{d u}{d v} 脳 \frac{d v}{d x}$ . Write this rule in 鈥減rime鈥� notation.

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = {(5 {(3 x^{4} - 1)}^{3} + 3 x - 1)}^{100}$

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the $z 鈫� x$ definition of the derivative

b) by using the $h 鈫� 0$ definition of the derivative

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

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