Explanation Given: The given function is f ( x ) = x Formula used: We have to use the Power Rule here. Power Rule: ( x n ) ' = n x n − 1 Calculation: We can write the given function as f ( x ) = x = x 1 2 We can find the first five derivatives of the given equation using the Power Rule. Therefore, we get f ' ( x ) = 1 2 x 1 2 − 1 = 1 2 x − 1 2 f ' ' ( x ) = − 1 2 × 1 2 x − 1 2 − 1 = − 1 4 x − 3 2 f ' ' ' ( x ) = − 3 2 × − 1 4 x − 3 2 − 1 = 3 8 x − 5 2 f ( 4 ) ( x ) = − 5 2 × 3 8 x − 5 2 − 1 = − 15 16 x − 7 2 f ( 5 ) ( x ) = − 15 16 × − 7 2 x − 7 2 − 1 = 105 To determine (b) The first five derivatives of f ( x ) = x and then show that f ( n ) ( x ) alternates in sign as − 1 ( n − 1 ) for n ≥ 1 To determine (c) The first five derivatives of f ( x ) = x and then find a formula for f ( n ) ( x ) for n ≥ 2

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 3.5, Problem 30E

To determine

(a)

The first five derivatives of f(x)=x and then show that

f(n)(x) is a multiple of x−n+12

To determine

(b)

The first five derivatives of f(x)=x and then show that f(n)(x) alternates in sign as −1(n−1) for n≥1

To determine

(c)

The first five derivatives of f(x)=x and then find a formula for f(n)(x) for n≥2

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Chapter 3.5, Problem 29E

Chapter 3.5, Problem 31E

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Chapter 3 Solutions

Calculus - Standalone book

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - Prob. 49E Ch. 3.1 - Prob. 50E Ch. 3.1 - Prob. 51E Ch. 3.1 - Prob. 52E Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 54E Ch. 3.1 - Prob. 55E Ch. 3.1 - 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Prob. 60CRE Ch. 3 - Prob. 61CRE Ch. 3 - Prob. 62CRE Ch. 3 - Prob. 63CRE Ch. 3 - Prob. 64CRE Ch. 3 - Prob. 65CRE Ch. 3 - Prob. 66CRE Ch. 3 - Prob. 67CRE Ch. 3 - Prob. 68CRE Ch. 3 - Prob. 69CRE Ch. 3 - Prob. 70CRE Ch. 3 - Prob. 71CRE Ch. 3 - Prob. 72CRE Ch. 3 - Prob. 73CRE Ch. 3 - Prob. 74CRE Ch. 3 - Prob. 75CRE Ch. 3 - Prob. 76CRE Ch. 3 - Prob. 77CRE Ch. 3 - Prob. 78CRE Ch. 3 - Prob. 79CRE Ch. 3 - Prob. 80CRE Ch. 3 - Prob. 81CRE Ch. 3 - Prob. 82CRE Ch. 3 - Prob. 83CRE Ch. 3 - Prob. 84CRE Ch. 3 - Prob. 85CRE

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(a) The first five derivatives of f ( x ) = x and then show that f ( n ) ( x ) is a multiple of x − n + 1 2

Solution Summary: The author explains how to determine the first five derivatives of f(x)=sqrtx using the Power Rule.

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(a)

The first five derivatives of f(x)=x and then show that

f(n)(x) is a multiple of x−n+12

(b)

The first five derivatives of f(x)=x and then show that f(n)(x) alternates in sign as −1(n−1) for n≥1

(c)

The first five derivatives of f(x)=x and then find a formula for f(n)(x) for n≥2

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Chapter 3 Solutions

(a) The first five derivatives of f ( x ) = x and then show that f ( n ) ( x ) is a multiple of x − n + 1 2

Solution Summary: The author explains how to determine the first five derivatives of f(x)=sqrtx using the Power Rule.

Concept explainers

Videos

(a) The first five derivatives of f(x)=x and then show that f(n)(x) is a multiple of x−n+12

(b) The first five derivatives of f(x)=x and then show that f(n)(x) alternates in sign as −1(n−1) for n≥1

(c) The first five derivatives of f(x)=x and then find a formula for f(n)(x) for n≥2

Want to see the full answer?

Chapter 3 Solutions

(a)

The first five derivatives of f(x)=x and then show that

f(n)(x) is a multiple of x−n+12

(b)

The first five derivatives of f(x)=x and then show that f(n)(x) alternates in sign as −1(n−1) for n≥1

(c)

The first five derivatives of f(x)=x and then find a formula for f(n)(x) for n≥2