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Use (a) the h→0 definition of the derivative and then (b) the z→c definition of the derivative to find f'(c) for each function f and value x = c.

f(x)=x23xx+1,x=0

Short Answer

Expert verified

(a) f'(c)=-3

(b)f'(c)=-3

Step by step solution

01

Part (a) Step 1. Given information.

Given function is f(x)=x23xx+1

We have to findf'(c)atx=0

02

Part (a) Step 2. Find the f'(c)

We have to find the derivative of the function using h→0 definition,

Therefore,

limh0f(0+h)f(0)h=limh0(0+h)23(0+h)0+h+1000+1h=limh0h23hh+1h=limh0h(h3)h(h+1)=limh0h(h3)h(h+1)=limh0h3h+1=3

03

Part (b) Step 1. Find f'(c)

Find the derivate of the function using x0definition,

limx0f(x)f(0)x0=limx0x23xx+1000+1x=limx0x(x3)x(x+1)=limx0x3x+1=3

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