Implicit differentiation question of the second order

Question

I’m trying to find $d^2y/dx^2$ in terms of $x$ and $y$ of $3xy-y^2=10$.

I’ve found that $y’=\frac{-3y}{2y+3x}$, but I’m having trouble finding $y’’$ Using implicit differentiation.

Answer 1

for the first derivative we get $$3y+3xy'-2yy'=0$$ or $$y'=\frac{3y}{2y-3x}$$ and for the second derivative: $$y''=\frac{3y'(2y-3x)-3y(2y'-3)}{(2y-3x)^2}$$ and here you can plug in $$y'=...$$ and this is equal to $$y''=\frac{18y^2-54xy}{(2y-3x)^3}$$