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Chapter 3: Q. 41 (page 239)

Use the definition of the derivative to find f' for each function f.

f(x)=x3+2

Short Answer

Expert verified

f'(x)=3x2

Step by step solution

01

Step 1. Given information. 

We have been given a function:

f(x)=x3+2

We have to find f' using the definition of the derivative.

02

Step 2. Find the derivative 

Derivative of the function can be found as :

limh鈫�0鈥�f(x+h)鈭�f(x)h=limh鈫�0鈥�(x+h)2+2鈭�x3+2h=limh鈫�0鈥�x3+3x2h+3xh2+h3+2鈭�x3鈭�2h=limh鈫�0鈥�h3x2+3xh+h2h=limh鈫�0鈥�3x2+3xh+h2=3x2

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Most popular questions from this chapter

Sketch the graph of a function f with the following properties:

f is continuous and defined on R;

f has critical points at x = 鈭�3, 0, and 5;

f has inflection points at x = 鈭�3, 鈭�1, and 2.

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=1-x4-2

Determine the graph of a function f from the graph of its derivative f'.

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f(x)=sinx,[a,b]=[0,蟺2].

Find the possibility graph of its derivative f'.
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