Problem

It is given that

$$x = 1 + \frac{1}{t} \qquad\text{and}\qquad y = \cos^{-1}t \qquad\text{for } 0 < t < 1.$$

(a) Show that

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

(b) Show that

$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = -t^a(1 - t^2)^b(2 - t^2),$$

where $a$ and $b$ are constants to be determined.