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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.

29.f(x)=x12,x=9

Short Answer

Expert verified

f'(9)=16.

Step by step solution

01

Part (a) Step 1. Given information. 

A function is given asf(x)=x12andx=c=9.

02

Part (a) Step 2. Solve f'(9) using h→0 definition of the derivative.

We have

f'(c)=limh0f(c+h)-f(c)hf'(9)=limh0f(9+h)-f(9)h=limh0(9+h)12-912h=limh09+h-9h=limh09+h-9h×9+h+99+h+9=limh09+h-9h(9+h+9)=limh0hh(9+h+9)=limh019+h+9=19+0+3=13+3=16

03

Part (b) Step 1. Solve f'(9) using z→9 definition of the derivative.

We have

f'(c)=limzcf(z)-f(c)z-cf'(9)=limz9f(z)-f(9)z-9=limz9z-9z-9=limz9z-9(z)2-(9)2=limz9z-9(z-9)(z+9)=limz91z+9=limz91z+3=19+3=13+3=16

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