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.