Explanation Given: The function, y = cos ( x 2 ) . 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 ( cos x ) = − sin x Derivative of any constant C is zero, that is, d d x ( C ) = 0 Calculation: In order to find the derivatives of the function, use chain rule and power rule of derivatives and simplify further to get y ' = d d x ( cos ( x 2 ) ) = − sin ( x 2 ) ⋅ d d x ( x 2 ) = − sin ( x 2 ) ⋅ ( 2 x 2 − 1 ) = − sin ( x 2 ) ⋅ ( 2 x ) = − 2 x sin ( x 2 ) So, the first derivative is y ' = − 2 x sin ( x 2 )

15th Edition

ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Chapter 3.5, Problem 37E

To determine

The firstderivative y' and second derivative y'' of the function, y=cos(x2) .

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Chapter 3.5, Problem 36E

Chapter 3.5, Problem 38E

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The firstderivative y ' and second derivative y ' ' of the function, y = cos ( x 2 ) .

Solution Summary: The author explains the first and second derivatives of the function, y=mathrmcos(x2).

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The firstderivative y' and second derivative y'' of the function, y=cos(x2) .

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