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Chapter 2: Q. 85 (page 199)

Use theh鈫�0definition of the derivative to prove the power rule holds for positive integers powers

Short Answer

Expert verified

We prove the power rule holds for positive integers powers using the definition of derivative

Step by step solution

01

Given information

We are given a function asf(x)=xn, Where n is a positive integer

02

Find the derivative

We use definition of the derivative

We have,

limh鈫�0f(x+h)-f(x)hlimh鈫�0(x+h)n-xnhlimh鈫�0xn+nxn-1h+.....+hn-xnh[binomialexpansion]limh鈫�0nxn-1h+.....+hnhlimh鈫�0h(nxn-1+....+hn-1)hlimh鈫�0(nxn-1+....+hn-1)=nxn-1

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Most popular questions from this chapter

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=1-4x6;f(1)=3

Use the definition of the derivative to find f'for each function fin Exercises 34-59

f(x)=1x2

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

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Use the definition of the derivative to find f'for each function fin Exercises 34-59

f(x)=3x
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