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 32 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

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 3.5, Problem 30E

To determine

(a)

The first five derivatives of $f (x) = \sqrt{x}$ and then show that

$f^{(n)} (x)$ is a multiple of $x^{- n + \frac{1}{2}}$

To determine

(b)

The first five derivatives of $f (x) = \sqrt{x}$ and then show that $f^{(n)} (x)$ alternates in sign as $- 1^{(n - 1)}$ for $n \geq 1$

To determine

(c)

The first five derivatives of $f (x) = \sqrt{x}$ and then find a formula for $f^{(n)} (x)$ for $n \geq 2$

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Chapter 3.5, Problem 29E

Chapter 3.5, Problem 31E

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Chapter 3 Solutions

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

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

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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

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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 $f (x) = \sqrt{x}$ and then show that

$f^{(n)} (x)$ is a multiple of $x^{- n + \frac{1}{2}}$

(b)

The first five derivatives of $f (x) = \sqrt{x}$ and then show that $f^{(n)} (x)$ alternates in sign as $- 1^{(n - 1)}$ for $n \geq 1$

(c)

The first five derivatives of $f (x) = \sqrt{x}$ and then find a formula for $f^{(n)} (x)$ for $n \geq 2$