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Definition-of-derivative calculations: Use the definition of the derivative to find f ' for each function

f(x)=x4f(x)=x-2f(x)=x2(x+1)f(x)=x2x+1

Short Answer

Expert verified

f'(x)=4x3f'(x)=-2x3f'(x)=3x2+2xf'(x)=x2+2xx+12

Step by step solution

01

Given Information.

Consider the given information.

f(x)=x4f(x)=x-2f(x)=x2(x+1)f(x)=x2x+1

02

Find the derivative of fx=x4.

Definition of derivative formula

f'(x)=limh0f(x+h)-f(x)hf(x)=x4f(x+h)=(x+h)4=x4+4x3h+6x2h2+4xh3+h4f'(x)=limh0x4+4x3h+6x2h2+4xh3+h4-x4hf'(x)=limh04x3h+6x2h2+4xh3+h4hf'(x)=limh04x3+6x2h+4xh2+h3f'(x)=4x3

03

Find the derivative of fx=x-2.

Use the definition and find the derivative.

f(x)=x-2f(x+h)=(x+h)-2=1x+h2=1x2+2xh+h2f'(x)=limh01x2+2xh+h2-1x2hf'(x)=limh0x2-(x2+2xh+h2)h*x2x2+2xh+h2f'(x)=limh0x2-x2-2xh-h2h*x2x2+2xh+h2f'(x)=limh0-2x-hx2x2+2xh+h2f'(x)=-2x3
04

Find the derivative of fx=x2x+1.

Use the definition to find the derivative.

f'(x)=limh0fx+hfxhf'(x)=limh0x+h2x+h+1-x2x+1hf'(x)=limh0x3+h2+x2+3x2h+3xh2+2xh+h3-x3-x2hf'(x)=limh0h3+h2+3x2h+3xh2+2xhhf'(x)=limh0h2+h+3x2+3xh+2xf'(x)=3x2+2x
05

Find the derivative of fx=x2x+1.

Use the definition of derivative to find it.

f(x)=x2x+1f'(x)=limh0fx+hfxh=limh0x+h2x+h+1-x2x+1h=limh0xh2+x2h+2xh+h2x+1x+h+1h=limh0xh2+x2h+2xh+h2x+1x+h+1h=limh0xh+x2+2x+hx+1x+h+1f'(x)=x2+2xx+12

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