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Chapter 14: Q. 32 (page 897)

In Problems 21鈥�32, find the derivative of each function at the given number.

f(x)=cosxat0

Short Answer

Expert verified

The derivative of the given function at is equal to0

Step by step solution

01

Step 1. Given information

Given function is f(x)=cosx

We have to find the derivative of the given function at0

02

Step 2. Definition of the derivative

Let's say a function y=f(x),

The derivative of the function at cis defined as

f'(c)=limx鈫�cf(x)-f(c)x-c

03

Step 3. Finding the derivative

By substituting the given function in the derivative definition,

we will get:

f'(0)=limx鈫�0f(x)-f(0)x-0鈬�f'(0)=limx鈫�0cosx-cos0x鈬�f'(0)=limx鈫�0cosx-1x

04

Step 4. Simplifying the limit

The expansion of cosxis cosx=1-x22!+x44!-x66!+..........

Now the limit is:

f'(0)=limx鈫�0(1-x22!+x44!-x66!+.......)-1x鈬�f'(0)=lim(x鈫�0-x2!+x34!-x56!+.......)鈬�f'(0)=0

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