Hi there. Well, I wanted to know how to find the second derivatives of a function using implicit differentiation. Is it possible? I think it is. I think I must use the chain rule somehow, but I don't know how... I'm in multivariable calculus since the function I'm gonna use could be seen as a function of only one variable.

An ellipse: $\displaystyle F(x,y)=4x^2+y^2-25=0$

So we have the partial derivatives:

$\displaystyle F_x=8x$, $\displaystyle F_y=2y$

$\displaystyle F_{xx}=8$, $\displaystyle F_{yy}=2$

So then, using implicit differentiation:

$\displaystyle \frac{{\partial x}}{{\partial y}}=-\displaystyle\frac{\frac{{\partial F}}{{\partial y}}}{\frac{{\partial F}}{{\partial x}}}=\displaystyle\frac{-y}{4x}$

But now if I wanna find $\displaystyle \frac{{\partial^2 x}}{{\partial y^2}}$ how should I proceed?