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Use (a) the h0definition of the derivative and then

(b) the zcdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23–38.

24.f(x)=x3,x=1

Short Answer

Expert verified

f'(1)=3

Step by step solution

01

Part (a) Step 1. Given information.

A function is given asf(x)=x3andx=c=1.

02

Part (a) Step 2. Find f'(1) using h→0 definition of the derivative.

We have

f'(c)=limh0f(c+h)-f(c)hf'(1)=limh0f(1+h)-f(1)h=limh0(1+h)3-13h=limh0[(1+h)-1)][(1+h)2+(1+h)1+1)]h=limh0h(1+h2+2h+2+h)h=limh0(h2+3h+3)=02+3(0)+3=3

03

Part (b) Step 2. Find f'(1) using z→1 definition of the derivative.

We have

f'(c)=limzcf(z)-f(c)z-cf'(1)=limz1f(z)-f(1)z-1=limz1(z)3-13z-1=limz1(z-1)(z2+z+1)z-1=limz1(z2+z+1)=12+1+1=1+1+1=3

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Most popular questions from this chapter

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