Explanation Proof: Consider a function f ( x ) = u ( x ) v ( x ) . The derivative of the function is f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h (1) Substitute x = x + h in the equation f ( x ) = u ( x ) v ( x ) , f ( x + h ) = u ( x + h ) v ( x + h ) Substitute the values f ( x ) = u ( x ) v ( x ) and f ( x + h ) = u ( x + h ) v ( x + h ) in the equation (1), f ′ ( x ) = lim h → 0 u ( x + h ) v ( x + h ) − u ( x ) v ( x ) h = lim h → 0 u ( x + h ) v ( x ) − u ( x ) v ( x + h ) v ( x + h ) ( v ( x ) ) ( h ) Add and subtract u ( x ) v ( x ) in the numerator. f ′ ( x ) = lim h → 0 u ( x + h ) v ( x ) − u ( x ) v ( x ) + u ( x ) v ( x ) − u ( x ) v ( x + h ) v ( x + h ) ( v ( x ) ) ( h ) Take v ( x ) as common from the first two terms and u ( x ) as common from the last two terms. f ′ ( x ) = lim h → 0 v ( x ) [ u ( x + h ) − u ( x ) ] − u ( x ) [ − v ( x ) + v ( x + h ) ] v ( x + h ) ( v ( x ) ) ( h ) On further simplification, f ′ ( x ) = lim h → 0 v ( x ) [ u ( x + h ) − u ( x ) ] v ( x + h ) ( v ( x ) ) ( h ) −

Calculus with Applications Books a la Carte Edition

11th Edition

ISBN: 9780133864564

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

Publisher: Pearson Education

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Chapter 4.2, Problem 37E

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To prove: The quotient rule for the derivatives.

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Chapter 4.2, Problem 36E

Chapter 4.2, Problem 38E

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Chapter 4 Solutions

Calculus with Applications Books a la Carte 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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