Problem

The curve $C$ has equation

$$4y^2 + 4\ln(xy) = 1.$$

(a) Show that, at the point $\left(2, \frac{1}{2}\right)$ on $C$,

$$\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{1}{6}.$$

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