Explanation Given information: The function is f ( x ) = 3 x x 2 − 2 and the point is ( 2 , 3 ) . Formula used : 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 . Calculation: The derivative of the given function gives the slope of the tangent line. Apply the product rule of differentiation in the given function to get the derivative function f ′ ( x ) . f ′ ( x ) = d d x [ 3 x x 2 − 2 ] = ( x 2 − 2 ) ⋅ d d x ( 3 x ) − ( 3 x ) ⋅ d d x ( x 2 − 2 ) ( x 2 − 2 ) 2 = ( x 2 − 2 ) ⋅ ( 3 ) − ( 3 x ) ⋅ ( 2 x ) ( x 2 − 2 ) 2 = 3 x 2 − 6 − 6 x 2 ( x 2 − 2 ) 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x ) . f ′ ( x ) = − 3 x 2 − 6 ( x 2 − 2 ) 2 ( Combine like terms ) (1) The slope of tangent line is, m = f ′ ( x ) Substitute 1 for x in the above equation to get the slope of tangent line from the point ( 2 , 3 )

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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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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Chapter 3.2, Problem 46E

To determine

To find: The equation of the tangent line to the graph of the given function f(x) at the given point.

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Chapter 3.2, Problem 45E

Chapter 3.2, Problem 47E

Chapter 3 Solutions

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The equation of the tangent line to the graph of the given function f ( x ) at the given point.

Solution Summary: The author explains how to find the equation of the tangent line to the graph of a given function f(x).

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To find: The equation of the tangent line to the graph of the given function f(x) at the given point.

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Chapter 3 Solutions