Explanation Given: f ( x ) = a ( n − x ) n and g ( x ) = b x n for n ≠ 0 . Result used: Quotient Rule ( a ( x ) b ( x ) ) ′ = a ′ ( x ) b ( x ) − a ( x ) b ′ ( x ) b ( x ) 2 Calculation: Substitute f ( x ) = a ( n − x ) n and g ( x ) = b x n in [ f ( x ) g ( x ) ] and simplify it as follows. [ ( a ( n − x ) n ) ( b x n ) ] = a b ( x n ( n − x ) n ) = a b ( x ( n − x ) ) n Obtain the derivative of [ f ( x ) g ( x ) ] as follows. [ a b ( x ( n − x ) ) n ] ′ = a b n ( x ( n − x ) ) n − 1 ( x ( n − x ) ) ′ = a b n ( x ( n − x ) ) n − 1 ( ( x ) ′ ( n − x ) − x ( n − x ) ′ ( n − x ) 2 ) [ d d x ( x n ) = n x n − 1 and by quotient rule ] = a b n x n − 1 ( n − x ) n − 1 ( 1 ( n − x ) − x ( 0 − 1 ) ( n − x ) 2 ) = a b n x n − 1 ( n − x ) n − 1 + 2 ( n − x + x ) On further simplification, [ ( a ( n − x ) n ) ( b x n ) ] ′ = a b n 2 x n − 1 ( n − x ) n + 1

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 4, Problem 67RE

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To calculate: The value of [f(x)g(x)]′ and f′(x)g′(x) and to verify there equivalence.

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Chapter 4, Problem 66RE

Chapter 4, Problem 68RE

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Chapter 4 Solutions

Calculus with Applications (11th Edition)

Ch. 4.1 - YOUR TURN 1 find f′(t). Ch. 4.1 - YOUR TURN 2 find dy/dx. Ch. 4.1 - Prob. 3YT Ch. 4.1 - Prob. 4YT Ch. 4.1 - Prob. 5YT Ch. 4.1 - Prob. 1WE Ch. 4.1 - Prob. 2WE Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...Ch. 4.1 - Find the derivative of each function defined as...

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The value of [ f ( x ) g ( x ) ] ′ and f ′ ( x ) g ′ ( x ) and to verify there equivalence.

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To calculate: The value of [f(x)g(x)]′ and f′(x)g′(x) and to verify there equivalence.

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Chapter 4 Solutions