Answer
$-\frac{\pi}{4}$
Work Step by Step
$(fog)'(x)$ = $f'(g(x))g'(x)$
$g(x)$ = $5\sqrt x$
$g'(x)$ = $\frac{5}{2\sqrt x}$
$g'(1)$ = $\frac{5}{2\sqrt 1}$ = $\frac{5}{2}$
$f(u)$ = $cot(\frac{\pi(u)}{10})$
$f'(u)$ = $-csc^{2}(\frac{\pi(u)}{10})[\frac{\pi}{10}]$
$f'(u)$ = $-\frac{\pi}{10}csc^{2}(\frac{\pi(u)}{10})$
$f'(u)$ = $f'(g(x))$ = $-\frac{\pi}{10}csc^{2}(\frac{5\pi\sqrt x}{10})$ = $-\frac{\pi}{10}csc^{2}(\frac{\pi\sqrt x}{2})$
$f'(g(1))$ = $-\frac{\pi}{10}csc^{2}(\frac{\pi\sqrt 1}{2})$ = $-\frac{\pi}{10}csc^{2}(\frac{\pi}{2})$ = $-\frac{\pi}{10}(1)$ = $-\frac{\pi}{10}$
$(fog)'(1)$ = $f'(g(1))g'(1)$ = $(-\frac{\pi}{10})$$(\frac{5}{2})$ = $-\frac{\pi}{4}$
