Problem

The curve $C$ has equation

$$x^3+2xy+8y^3=-12.$$

(a) Show that, at the point $(-2,-1)$ on $C$,

$$\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{2}.$$

(b) Find the value of $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ at the point $(-2,-1)$.