Chapter 2: Q. 88 (page 235)

Use implicit differentiation and the fact that $s e c (s e c^{- 1} x) = x$ for all $x$ in the domain of $s e c^{- 1} x$ to prove that $\frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}$ . You will have to consider the cases $x > 1$ and $x < - 1$ separately.

Short Answer

Expert verified

We proved $\frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}$ usingimplicit differentiation.

Step by step solution

Step 1. Given Information

We are given $s e c^{- 1} x$ and using implicit differentiation we need to prove that $\frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}$ .

Step 2. Proving the expression

We consider $x > 1$ and $x < - 1$ ,

$s e c (s e c^{- 1} x) = x$

Differentiating both sides we get,

\frac{d}{d x} \{s e c (s e c^{- 1} x)\} = \frac{d}{d x} (x) \Rightarrow s e c (s e c^{- 1} x) \tan (s e c^{- 1} x) \frac{d}{d x} (s e c^{- 1} x) = 1 \Rightarrow \frac{d}{d x} (s e c^{- 1} x) = \frac{1}{s e c (s e c^{- 1} x) \tan (s e c^{- 1} x)} \Rightarrow \frac{d}{d x} (s e c^{- 1} x) = \frac{1}{x t an (s e c^{- 1} x)} \Rightarrow \frac{d}{d x} (s e c^{- 1} x) = \frac{1}{x (|x| \sqrt{1 - \frac{1}{{(s e c (s e c^{- 1} x))}^{2}}})} \Rightarrow \frac{d}{d x} (s e c^{- 1} x) = \frac{1}{x (|x| \frac{\sqrt{x^{2} - 1}}{x})} \Rightarrow \frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}

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Use implicit differentiation and the fact that $s e c (s e c^{- 1} x) = x$ for all $x$ in the domain of $s e c^{- 1} x$ to prove that $\frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}$ . You will have to consider the cases $x > 1$ and $x < - 1$ separately.

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 secsec-1x=xfor all xin the domain of sec-1xto prove that ddxsec-1x=1xx2-1. You will have to consider the casesx>1andx<-1 separately.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the expression

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Use implicit differentiation and the fact that $s e c (s e c^{- 1} x) = x$ for all $x$ in the domain of $s e c^{- 1} x$ to prove that $\frac{d}{d x} (s e c^{- 1} x) = \frac{1}{|x| \sqrt{x^{2} - 1}}$ . You will have to consider the cases $x > 1$ and $x < - 1$ separately.