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Chapter 2: Q. 42 (page 184)

Use the definition of the derivative to find f' for each function f in Exercises 39-54

f(x)=-x3-x+1

Short Answer

Expert verified

The value off'(x)=-3x2+1

Step by step solution

01

Step 1. Given information

The given functionf(x)=-x3-x+1

02

Step 2. Finding the value of f'(x)

We know that f'(x)=limh鈫�0f(x+h)-f(x)h.........(1)

Given f(x)=-x3-x+1then

f(x+h)=-(x+h)3-(x+h)+1

Putting the values in (1)

f'(x)=limh鈫�0(-(x+h)3-(x+h)+1)-(-x3-x+1)h=limh鈫�0(-(x3+h3+3x2h+3xh2)-(x+h)+1))+x3+x-1h[鈭�(a+b)3=a3+b3+3a2b+3ab2]=limh鈫�0-x3-h3-3x2h-3xh2-x-h+x3+x-1h=limh鈫�0-h3-3x2h-3xh2+hh=limh鈫�0h(-h2-3x2-3xh+1)h=limh鈫�0-h2-3x2-3xh+1

Putting h=0

=-(0)2-3x2-3x(0)+1=-3x2+1

Hence,f'(x)=-3x2+1



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