If x = f ( t ) and y = g ( t ) are differentiable functions of t then d 2 y d x 2 is

Question

If x = f ( t ) and y = g ( t ) are differentiable functions of t then d 2 y d x 2 is, f ′ ( t ) ⋅ g ″ ( t ) − g ′ ( t ) ⋅ f ″ ( t ) [ f ′ ( t ) ] 3, f ′ ( t ) ⋅ g ″ ( t ) − g ′ ( t ) ⋅ f ″ ( t ) [ f ′ ( t ) ] 2, g ′ ( t ) ⋅ f ″ ( t ) − f ′ ( t ) ⋅ g ″ ( t ) [ f ′ ( t ) ] 3, g ′ ( t ) ⋅ f ″ ( t ) + f ′ ( t ) ⋅ g ″ ( t ) [ f ′ ( t ) ] 3

MARKS App · Accepted Answer

X = f ( t ) Y = g ( t ) d x d t = f ′ ( t ) d x d t = g ′ ( t ) d y d x = d y d t d x d t = g ′ ( t ) f ′ ( t ) d 2 y d x 2 = d d x ( g ′ ( t ) f ′ ( t ) ) = d d t ( g ′ ( t ) g ′ ( t ) ) . d t d x = f ′ ( t ) ⋅ g ″ ( t ) − g ′ ( t ) ⋅ f ″ ( t ) ( f ′ ( t ) ) 2 ⋅ 1 f ′ ( t ) = f ′ ( t ) ⋅ g ″ ( t ) − g ′ ( t ) ⋅ f ″ ( t ) ( f ′ ( t ) ) 3

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