Implicit differentiation and inverse trigonometric functions practice question 27

You need $y'$ first, then a second differentiation. Both $y$ and $y'$ change with $x$, so when is it safe to put in the point?

First $3x^{2} + 3y^{2}\,y' = 0$, so $y' = -\dfrac{x^{2}}{y^{2}}$. Find its value at $(-1, 2)$.

Differentiate $x^{2} + y^{2}\,y' = 0$ once more: $y^{2}\,y'$ is a product, and the derivative of $y^{2}$ is $2y\,y'$. Substitute the point and the value of $y'$ only after that.