Explanation Given information: The derivative exists. Chain rule. Let g, f and u are the functions of x, then d f d x = d f d u ⋅ d u d x Calculation: Let u ( x ) = f ( x ) and g ( x ) = f ' ( x ) . Then, f ' [ f ( x ) ] = f ' ( u ( x ) ) f ' [ f ( x ) ] = g ( u ( x ) ) From chain rule. d d x f ' ( f ( x ) ) = d g ( u ) d u ⋅ d u ( x ) d x d d x f ' ( f ( x ) ) = d f ' ( u ) d u ⋅ d f ( x ) d x [ substitute g ( x ) = f ' ( u ) and u ( x ) = f ( x ) ] d d x f ' ( f ( x ) ) = f ' ' ( u ) ⋅ f ' ( x ) d d x f ' ( f ( x ) ) = f ' ' ( f ( x ) ) ⋅ f ' ( x ) [ substitute u = f ( x ) ] Thus, the derivativeis d d x f ' ( f ( x ) ) = f ' ' ( f ( x ) ) ⋅ f ' ( x )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3.5, Problem 59PS

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To find the derivative of function ddxf'[f(x)] and ddxf[f'(x)] .

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Chapter 3.5, Problem 58PS

Chapter 3.5, Problem 60PS

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Chapter 3 Solutions

Calculus

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To find the derivative of function d d x f ' [ f ( x ) ] and d d x f [ f ' ( x ) ] .

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To find the derivative of function ddxf'[f(x)] and ddxf[f'(x)] .

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Chapter 3 Solutions