Explanation Given information: The given function is f ( x ) = 1 + 2 x 1 / 2 1 + x 3 / 2 and the point is ( 4 , 5 9 ) . Formula used : (1) The quotient rule of differentiation is, f ′ ( x ) = d d x [ f ( x ) g ( x ) ] = g ( x ) ⋅ d d x f ( x ) − f ( x ) ⋅ d d x g ( x ) [ g ( x ) ] 2 The above formula is defined when the denominator g ( x ) ≠ 0 . (2) The point-slope form of equation is, y − y 1 = m ( x − x 1 ) (1) In the formula, m is the slope of line and x 1 , y 1 , x 2 and y 2 are coordinates of two points. Calculation: Section1: The derivative of the given function gives the slope of the tangent line. Apply the quotient rule of differentiation for the given function to get the derivative function f ′ ( x ) . f ′ ( x ) = d d x [ 1 + 2 x 1 / 2 1 + x 3 / 2 ] = ( 1 + x 3 / 2 ) ⋅ d d x ( 1 + 2 x 1 / 2 ) − ( 1 + 2 x 1 / 2 ) ⋅ d d x ( 1 + x 3 / 2 ) ( 1 + x 3 / 2 ) 2 = ( 1 + x 3 / 2 ) ⋅ ( 2 ⋅ 1 2 ⋅ x − 1 / 2 ) − ( 1 + 2 x 1 / 2 ) ⋅ ( 3 2 x 1 / 2 ) ( 1 + x 3 / 2 ) 2 = x − 1 2 ( 1 + x 3 2 ) − 3 2 x 1 2 ( 1 + 2 x 1 2 ) ( 1 + x 3 2 ) 2 ( Apply distributive property ) (1) The slope of tangent line is, m = f ′ ( x ) Substitute 4 for x above equation to get the slope of tangent line from the point ( 4 , 5 9 ) . m = f ′ ( 4 ) (2) Substitute 4 for x in equation (1) to get the value of f ′ ( 4 )

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 3.2, Problem 42E

To determine

To find: The slope and an equation of the tangent line of the given function that passes through a given point.

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

Ch. 3.1 - State the following rules of differentiation in...Ch. 3.1 - Prob. 2CQ Ch. 3.1 - Prob. 3CQ Ch. 3.1 - Prob. 4CQ 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 Ch. 3.1 - Prob. 6E

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The slope and an equation of the tangent line of the given function that passes through a given point.

Solution Summary: The author explains the quotient rule of differentiation for the given function to get the derivative function.

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To find: The slope and an equation of the tangent line of the given function that passes through a given point.

Chapter 3 Solutions