How do I use implicit function theorem to get second derivative of a function?

Question

Consider

$x^2 + y^2 = r^2$

acc. to implicit function theorem,

$ y' = \frac{-x}{y}$

Now how would I find y'' by using implicit theorem again?

and what would be my multivariable function? , would it be $F(x,y,y') = y y' +x $

EDit: I mean using implicit function theorem again... I don't want direct derivative!!

Answer 1

You have

$$y y^\prime +x = 0.$$ Differentiate this relation according to $x$, you get

$$\left(y^\prime\right)^2 + y y^{\prime \prime} + 1 =0.$$

Replace $y' = \frac{-x}{y}$ in the equation above and you get:

$$y^{\prime \prime} = - \frac{1}{y} - \frac{x^2}{y^3}$$