Second Derivative of an implicit function using partial differentiation

Question

If $x\sqrt{1-y^2} + y\sqrt{1-x^2} = a$

I have to show that:

$\frac{d^2{y}}{d{x^2}} = \frac{-a}{(1-x^2)^\frac{3}{2}}$

What I did is, used the formula :

$\mathbf{\frac{dy}{dx} = -\frac{\frac{\partial{f}}{\partial{x}}}{\frac{\partial{f}}{\partial{y}}}}$

Using the above formula, I calculated,

$\frac{dy}{dx}= - \frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$

However for the second derivative, I just need to differentiate the above equation, but I am having a hard time to prove it, Is there any way to sort it out?

Answer 1

If you have $$y'= - \frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$$ differentiate again to obtain $$y''=\frac{y y'}{\sqrt{1-x^2} \sqrt{1-y^2}}-\frac{x \sqrt{1-y^2}}{\left(1-x^2\right)^{3/2}}$$ Replace or not $y'$ by its expression.