4. (i)
$$f ( x ) = x \arccos x \quad - 1 \leqslant x \leqslant 1$$
Find the exact value of \(f ^ { \prime } ( 0.5 )\).
(ii)
$$\mathrm { g } ( x ) = \arctan \left( \mathrm { e } ^ { 2 x } \right)$$
Show that
$$\mathrm { g } ^ { \prime \prime } ( x ) = k \operatorname { sech } ( 2 x ) \tanh ( 2 x )$$
where \(k\) is a constant to be found.
\includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-15_2647_1840_119_114}
- Prove that for \(n \geqslant 2\)
$$( n - 1 ) I _ { n } = \tan x \sec ^ { n - 2 } x + ( n - 2 ) I _ { n - 2 }$$
- Hence, showing each step of your working, find the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 6 } x d x$$
$$I _ { n } = \int \sec ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$
Prove that for \(n > 2\)