Q. 88 Use implicit differentiation and... [FREE SOLUTION]

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

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91影视

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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91影视

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Discrete Mathematics

Statistics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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.