Problem

Find the $ n $ th derivative of each function by …

02:31

Problem 65 Easy Difficulty

Use the definition of a derivative to show that if $ f(x) = 1/x, $ then $ f'(x) = -1/x^2. $ (This proves the Power Rule for the case $ n = -1.$ )

Answer

$$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\lim _{h \rightarrow 0} \frac{x-(x+h)}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-h}{h x(x+h)}=\lim _{h \rightarrow 0} \frac{-1}{x(x+h)}=-\frac{1}{x^{2}}$$

More Answers

01:01

Frank L.

Topics

Derivatives

Differentiation

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Discussion

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Watch More Solved Questions in Chapter 3

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Video Transcript

Hey, it's clear, So enumerated here. So we have up of access equal Thio one over X. Yeah, if the door If it is, it's equal to limit. As each approaches Ciro one over X Plus H minus one over X over each becomes equal to the limit knows each approaches Ciro for a negative H over X X plus H over each. This is equal to limit as each approaches. So for negative one over X terms X plus H, which is equal to negative one over X square.

Clarissa N.

Topics

Derivatives

Differentiation

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Top Calculus 1 / AB Educators

Caleb E.

Baylor University

Kristen K.

University of Michigan - Ann Arbor

Samuel H.

University of Nottingham

Michael J.

Idaho State University