Explanation Given information: The given function is f ( x ) = x + 1 x 2 + 1 and the point is ( 1 , 1 ) . 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 [ x + 1 x 2 + 1 ] = ( x 2 + 1 ) ⋅ d d x ( x + 1 ) − ( x + 1 ) ⋅ d d x ( x 2 + 1 ) ( x 2 + 1 ) 2 = ( x 2 + 1 ) ⋅ ( 1 ) − ( x + 1 ) ⋅ ( 2 x ) ( x 2 + 1 ) 2 = x 2 + 1 − 2 x 2 − 2 x ( x 2 + 1 ) 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x )

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.2, Problem 41E

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 3.2, Problem 40E

Chapter 3.2, Problem 42E

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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.

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Chapter 3 Solutions