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Chapter 14: Q. 31 (page 897)

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

f(x)=sinxat0

Short Answer

Expert verified

The derivative of the given function at 0is equal to1.

Step by step solution

01

Step 1. Given information

Given function is f(x)=sinx

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 c is 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�0sinx-sin0x鈬�f'(0)=limx鈫�0sinxx

04

Step 4. Simplifying the limit

The expansion of sinxis:

sinx=x-x33!+x55!-x77!+........

Now the limit is:

localid="1647086780847">f'(0)=limx鈫�0x-x33!+x55!-x77!+.......x鈬�f'(0)=limx鈫�01-x23!+x45!-x67!+.......鈬�f'(0)=1

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Most popular questions from this chapter

True or False The exponential function f(x)=ex is increasing on the interval (-鈭�,鈭�). (p. 280)

True or False For any function f, and any real number c,limx鈫�c-f(x)=limx鈫�c+f(x).

In Problems 53-56, use the properties of limits and the facts that

limx鈫�0sinxx=1limx鈫�0cosx-1x=0limx鈫�0sinx=0limx鈫�0cosx=1

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