Explanation Given information: The given function is g ( t ) = t 2 t − 1 . 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 ) ] by Quotient Rule is, h ′ ( x ) = f ( x ) g ′ ( x ) − f ′ ( x ) g ( x ) f 2 ( x ) Calculation: The given function is, g ( t ) = t 2 t − 1 (1) Differentiate equation (1) with respect to t on both sides to evaluate g ′ ( t ) . g ′ ( t ) = ( t − 1 ) d d t ( t 2 ) − t 2 d d t ( t − 1 ) ( t − 1 ) 2 [ Use Quotient Rule ] = 2 t ( t − 1 ) − t 2 ( t − 1 ) 2 [ Use General Power Rule d d x ( x n ) = n x n − 1 , n > 0 ] = 2 t 2 − 2 t − t 2 ( t − 1 ) 2 = t 2 − 2 t ( t − 1 ) 2 Again differentiate above equation with respect to t on the both sides, to evaluate g ″ ( t )

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Differential Equations

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Chapter 3.5, Problem 16E

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

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Chapter 3.5, Problem 15E

Chapter 3.5, Problem 17E

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The first and second derivative of given function.

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