Problem

(i) $f(x) = x\arccos x \qquad -1 \le x \le 1$

Find the exact value of $f'(0.5)$.

(ii) $g(x) = \arctan(e^{2x})$

Show that

$$g''(x) = k\,\operatorname{sech}(2x)\tanh(2x)$$

where $k$ is a constant to be found.