Explanation Given information: The figures as shown below. Formula used: The chain rule, d d x h ( x ) = d d x ( f ( g ( x ) ) ) h ′ ( x ) = f ′ ( g ( x ) ) ⋅ g ′ ( x ) Calculation: Consider the function k ( x ) = g ( f ( x ) ) . Find the first derivative with respect to x as below. Thus, d d x ( k ( x ) ) = d d x ( g ( f ( x ) ) ) k ′ ( x ) = g ′ ( f ( x ) ) ⋅ f ′ ( x ) Substitute x = 1 into k ′ ( x ) = g ′ ( f ( x ) ) ⋅ f ′ ( x ) . Thus, k ′ ( x ) = g ′ ( f ( x ) ) ⋅ f ′ ( x ) k ′ ( 1 ) = g ′ ( f ( 1 ) ) ⋅ f ′ ( 1 ) f ( 1 ) = − 0.5 = g ′ ( − 0.5 ) ⋅ f ′ ( 1 ) … ( 1 ) Sketch lines on the graph as below. Consider the section of the graph as a straight line to estimate the slope. The slope of the line. m = y 2 − y 1 x 2 − x 1 = − 0.5 + 0.8 1 − 0 = 0

6th Edition

ISBN: 9781119399353

Author: DA

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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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 3.3, Problem 42P

To determine

To calculate:

The derivative k′(1) if k(x)=g(f(x)).

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Chapter 3.3, Problem 41P

Chapter 3.3, Problem 43P

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The derivative k ′ ( 1 ) if k ( x ) = g ( f ( x ) ) .

Solution Summary: The author explains how to calculate the derivative kprime(1)is1.5.

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To calculate:

The derivative k′(1) if k(x)=g(f(x)).

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Chapter 3 Solutions

The derivative k ′ ( 1 ) if k ( x ) = g ( f ( x ) ) .

Solution Summary: The author explains how to calculate the derivative kprime(1)is1.5.

Concept explainers

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To calculate: The derivative k′(1) if k(x)=g(f(x)).

Want to see the full answer?

Chapter 3 Solutions

To calculate:

The derivative k′(1) if k(x)=g(f(x)).