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Chapter 2: Q 2. (page 237)

Find the derivatives of f(x)=x3.

Short Answer

Expert verified

The derivative of f(x)=x3is3x2.

Step by step solution

01

Given Information

The given expression isf(x)=x3.

02

Simplification 

f'(x)=limh鈫�0f(x+h)-f(x)h=limh鈫�0(x+h)3-x3h=limh鈫�0x3+3x2h+3xh2+h3-x3h=limh鈫�0(3x2+3xh).hh=limh鈫�0(3x2+3xh)=3x2+3x.0=3x2

Therefore, the derivate off(x)=x3is3x2.

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Most popular questions from this chapter

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

25.f(x)=1x,x=-1

Use the definition of the derivative to find ffor each function fin Exercises 39-54.

f(x)=x2-1x2-x-2

Use the definition of the derivative to prove the following special case of the product rule

ddx(x2f(x))=2xf(x)+x2f'(x)

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x-12(x2-1)3

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x+1x2+1
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