Explanation Given information: The given function is g ( t ) = 3 t 2 ( 2 t 2 − 1 ) 2 . Formula used: The formula to calculate the differential of a function h ( x ) = [ g ( x ) ] n by General Power Rule is, h ′ ( x ) = n [ g ( x ) ] n − 1 ⋅ g ′ ( x ) The formula to calculate the differential of a function h ( x ) = [ g ( x ) f ( x ) k ( x ) ] by Product Rule is, h ′ ( x ) = f ( x ) k ( x ) g ′ ( x ) + f ′ ( x ) g ( x ) k ( x ) + f ( x ) g ( x ) k ′ ( x ) Calculation: The given function is, g ( t ) = 3 t 2 ( 2 t 2 − 1 ) 2 (1) Differentiate equation (1) with respect to t on both sides to evaluate g ′ ( t ) . g ′ ( t ) = g ( t ) = ( 2 t 2 − 1 ) 2 d d t ( 3 t 2 ) + 3 t 2 d d t ( 2 t 2 − 1 ) 2 [ Use Product Rule ] = 6 t ( 2 t 2 − 1 ) 2 + 3 t 2 ( 4 t ) 2 ( 2 t 2 − 1 ) [ Use General Power Rule d d x ( x n ) = n x n − 1 , n > 0 ] = 6 t ( 2 t 2 − 1 ) [ 2 t 2 − 1 + 4 t 2 ] = 6 t ( 2 t 2 − 1 ) [ 6 t 2 − 1 ] Again differentiate above equation with respect to t on the both sides, to evaluate g ″ ( t )

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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Differential Equations

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Chapter 3.5, Problem 9E

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To find: The first and second derivative of given function.

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Chapter 3.5, Problem 8E

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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 first and second derivative of given function.

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