Explanation Definition used: Derivative : “For the function y = f ( x ) , the derivative of f at x is given by, f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h if the limit exists”. Procedure used: The procedure of four-step process for finding the derivative of a function f as follows. Step 1: Find f ( x + h ) . Step 2: Find f ( x + h ) − f ( x ) . Step 3: Find f ( x + h ) − f ( x ) h . Step 4: Find lim h → 0 f ( x + h ) − f ( x ) h . Proof: Find the derivative of the function by using four-step process. Step 1: Find f ( x + h ) . Substitute x = x + h in the given function, f ( x + h ) = u ( x + h ) − v ( x + h ) = u ( x ) + u ( h ) − v ( x ) − v ( h ) Step 2: Find f ( x + h ) − f ( x ) . f ( x + h ) − f ( x ) = u ( x ) + u ( h ) − v ( x ) − v ( h ) − ( u ( x ) − v ( x ) ) = u ( x ) + u ( h ) − v ( x ) − v ( h ) − u ( x ) + v ( x ) = u ( h ) − v ( h ) Hence, f ( x + h ) − f ( x ) = u ( h ) − v ( h ) . Step 3: Find f ( x + h ) − f ( x ) h

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

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Chapter 2.5, Problem 88E

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To show: The derivative of the function f(x)=u(x)−v(x) is f′(x)=u′(x)−v′(x) by using the four-step process.

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Chapter 2.5, Problem 87E

Chapter 2.5, Problem 89E

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The derivative of the function f ( x ) = u ( x ) − v ( x ) is f ′ ( x ) = u ′ ( x ) − v ′ ( x ) by using the four-step process.

Solution Summary: The author explains the four-step process for finding the derivative of a function f.

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