Problem

$$y=\operatorname{arccos}(\operatorname{sech}x)\qquad x>0$$

(a) Show that

$$\frac{dy}{dx}=\operatorname{sech}x$$

Figure 1 shows a sketch of part of the curve $C$ with equation $y=f(x)$ where

$$f(x)=\operatorname{arccos}(\operatorname{sech}x)+\coth x \qquad x>0$$

The point $P$ is a minimum turning point of $C$

(b) Show that the $x$ coordinate of $P$ is $\ln(q+\sqrt q)$ where $q=\frac12(1+\sqrt{k})$ and $k$ is an integer to be determined.