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Chapter 14: Q. 29 (page 897)

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

f(x)=x3+x2-2xat1.

Short Answer

Expert verified

Derivative of the function at1 is equal to3

Step by step solution

01

Step 1. Given information

Given function isf(x)=x3+x2-2x

We have to find the derivative of the given function at1

02

Step 2. Definition of the derivative

Let's say a functiony=f(x)

The derivative of the function at cis defined as

localid="1647083580386" 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'(1)=limx鈫�1f(x)-f(1)x-1鈬�f'(1)=limx鈫�1(x3+x2-2x)-(13+12-2)x-1鈬�f'(1)=limx鈫�1x3+x2-2xx-1

04

Step 4. Simplifying the limit

By rewriting the limit, we will have:

f'(1)=limx鈫�1x3+x2-2xx-1鈬�f'(1)=limx鈫�1(x-1)(x2+2x)x-1鈬�f'(1)=limx鈫�1(x2+2x)鈬�f'(1)=12+2鈬�f'(1)=3

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