Problem

The curve $C$ has equation

$$9y^2 - 3\sinh^{-1}(xy) = 1 - 3\ln 3.$$

(a) Show that, at the point $\big(4, \frac{1}{3}\big)$ 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 $\big(4, \frac{1}{3}\big)$.