BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.3, Problem 18E
To determine

Find the derivative of the given function using definition of derivative.

Expert Solution

Explanation of Solution

Given:

The given expressionis f(x)=cosx .

Calculation:

From the definition of derivative.

  f'(x)=limh0f(x+h)f(x)hf(x)=cosxf(x+h)=cos(x+h)f'(x)=limh0cos(x+h)cosxh

Use formula cos(x+y)=cosxcosysinxsiny .

  f'(x)=limh0cosxcoshsinxsinhcosxhf'(x)=limh0cosxcoshcosxsinxsinhhf'(x)=limh0cosx(cosh1)sinxsinhhf'(x)=limh0cosx(cosh1h)sinx(sinhh)f'(x)=cosx(limh0cosh1h)sinx(limh0sinhh)

Use formulas

  limα0cosα1α=0limα0sinαα=1

  f'(x)=cosx0sinx1f'(x)=sinx

Hence f'(x)=sinx proved.

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