Q. 87 Use implicit differentiation and... [FREE SOLUTION]

Chapter 2: Q. 87 (page 235)

Use implicit differentiation and the fact that $\tan (\tan^{- 1} x) = x$ for all $x$ in the domain of $\tan^{- 1} x$ to prove that $\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$ .

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

We are given $\tan^{- 1} x$ and using implicit differentiation we need to prove that $\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$ .

Step 2. Proving the expression

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

Differentiating both sides we get,

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

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Use implicit differentiation and the fact that $\tan (\tan^{- 1} x) = x$ for all $x$ in the domain of $\tan^{- 1} x$ to prove that $\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the expression

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91影视

Use implicit differentiation and the fact that tantan-1x=xfor all xin the domain of tan-1x to prove that ddxtan-1x=11+x2 .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the expression

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Use implicit differentiation and the fact that $\tan (\tan^{- 1} x) = x$ for all $x$ in the domain of $\tan^{- 1} x$ to prove that $\frac{d}{d x} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$ .