Find y'' by implicit differentiation.

Question

Find y'' by implicit differentiation.

9x² + y² = 3

y"=

Answer 1

d/dx (9x²+y² -3 = 0) 18x + 2yy' = 0 or y' = -9x/y

Now take the implicit derivative of this (using quotient rule). You can sub in -9x/y for y' and sqrt(3-9x²) for y in your final expression for y''.

Answer 2

18x + 2yy' = 0

y' = -18x / (2y)

= (-9x)/y

y'' = [(-9y) - (-9xy')] / y^2 <--- quotient rule

= -9 ( xy'+y)/ y^2

Also y' = -9 x y^(-1)

Product rule says: -9 x y^(-2) y' + -9 y^(-1)

= -y^(-2) 9xy' + 9y

= -9( xy' + y)/y^2