Chapter 2: Q. 94 (page 236)

Prove each of the differentiation formulas in Exercises 93–96. (These exercises involve hyperbolic functions.)
$\frac{d}{d x} (\tan h x) = s e c h^{2} x$

Short Answer

Expert verified

We proved the formula $\frac{d}{d x} (\tan h x) = s e c h^{2} x$ .

Step by step solution

Step 1. Given Information

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

Step 2. Proving the function

$\frac{d}{d x} (\tan h x) = \frac{d}{d x} (\frac{\sin h x}{\cos h x}) = \frac{d}{d x} (\frac{\frac{e^{x} - e^{- x}}{2}}{\frac{e^{x} + e^{- x}}{2}}) = \frac{d}{d x} (\frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}) = \frac{\frac{d}{d x} (e^{x} - e^{- x}) (e^{x} + e^{- x}) - (e^{x} - e^{- x}) \frac{d}{d x} (e^{x} + e^{- x})}{{(e^{x} + e^{- x})}^{2}} = \frac{(e^{x} - (- e^{- x})) (e^{x} + e^{- x}) - (e^{x} - e^{- x}) (e^{x} - e^{- x})}{{(e^{x} + e^{- x})}^{2}} = \frac{(e^{x} + e^{- x}) (e^{x} + e^{- x}) - (e^{x} - e^{- x}) (e^{x} - e^{- x})}{{(e^{x} + e^{- x})}^{2}} = \frac{{(e^{x} + e^{- x})}^{2} - {(e^{x} - e^{- x})}^{2}}{{(e^{x} + e^{- x})}^{2}} = \frac{e^{2 x} + 2 e^{x} e^{- x} + e^{- 2 x} - e^{2 x} + 2 e^{2 x} e^{- x} - e^{- 2 x}}{{(e^{x} + e^{- x})}^{2}} = \frac{4 e^{x} e^{- x}}{{(e^{x} + e^{- x})}^{2}} = {(\frac{2}{(e^{x} + e^{- x})})}^{2} = \frac{1}{\cos h^{2} x} = s e c h^{2} x$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Prove each of the differentiation formulas in Exercises 93–96. (These exercises involve hyperbolic functions.)
$\frac{d}{d x} (\tan h x) = s e c h^{2} x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Geometry

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove each of the differentiation formulas in Exercises 93–96. (These exercises involve hyperbolic functions.)ddxtanhx=sech2x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Geometry

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove each of the differentiation formulas in Exercises 93–96. (These exercises involve hyperbolic functions.)
$\frac{d}{d x} (\tan h x) = s e c h^{2} x$