Explanation Given: The function, F ( w ) = e w ln ( w ) . Concept Used: The product rule of derivatives: d d x [ f ( x ) ⋅ g ( x ) ] = f ( x ) ⋅ d d x g ( x ) + g ( x ) ⋅ d d x f ( x ) Chain rule, d d x ( f ( g ( x ) ) ) = f ' ( g ( x ) ) ⋅ g ' ( x ) Power rule of derivatives, d d x ( x n ) = n x n − 1 d d x ( ln x ) = 1 x d d x ( e x ) = e x Derivative of any constant C is zero, that is, d d x ( C ) = 0 Calculation: In order to find the derivatives of the given function F ( w ) = e w ln ( w ) , use the product rule and chain rule of derivatives and simplify further as shown below: F ' ( w ) = d d w ( e w ln ( w ) ) = ( e w ) ⋅ d d w ( ln ( w ) ) + ln ( w ) ⋅ d d w ( e w ) = ( e w ) ⋅ ( 1 w ) + ln ( w ) ⋅ ( e w ) = e w w + e w ln ( w ) = e w ( 1 w + ln ( w ) ) So, the first derivative is F ' ( w ) = e w ( 1 w + ln ( w ) )

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

3rd Edition

ISBN: 9780840064189

Author: Frederick R. Adler

Publisher: Cengage Learning

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Chapter 2.8, Problem 13E

To determine

The firstand second derivativesof the function, F(w)=ewln(w) .

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Chapter 2.8, Problem 12E

Chapter 2.8, Problem 14E

Chapter 2 Solutions

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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The firstand second derivativesof the function, F ( w ) = e w ln ( w ) .

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The firstand second derivativesof the function, F(w)=ewln(w) .

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Chapter 2 Solutions