Q.382. Second order implicit differentiation.

Question 382.
Use implicit differentiation to find dy/dx and then d^(2)y/dx^2 of y^2 = 7x^(2) - 6x.

Answer 382.
y^2 = 7x^2 - 6x
=> 2y dy/dx = 14x - 6
=> dy/dx = (7x - 3) / y   ...   ( 1 )

Differentiating ( 1 ) wrt. x,
y d^2y/dx^2 + (dy/dx)^2 = 7
=> d^2y/dx^2
= [7 - (dy/dx)^2] / y
= [7 - (7x - 3)^2 / y^2] / y
= [7y^2 - (7x - 3)^2] / y^3
= [7 * (7x^2 - 6x) - (49x^2 - 42x + 9)] / y^3
= - 9/y^3.

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