Answer
$$y' = \frac{{5 - 4\sqrt \pi }}{{10}}$$
Work Step by Step
$$\eqalign{
& 5\sqrt x - 10\sqrt y = \sin x \cr
& {\text{Differentiate both sides with respect to }}x \cr
& 5\left( {\frac{1}{{2\sqrt x }}} \right) - 10\left( {\frac{1}{{2\sqrt y }}} \right)y' = \cos x \cr
& \frac{5}{{2\sqrt x }} - \frac{5}{{2\sqrt y }}y' = \cos x \cr
& \frac{5}{{2\sqrt y }}y' = \frac{5}{{2\sqrt x }} - \cos x \cr
& y' = \frac{{2\sqrt y }}{5}\left( {\frac{5}{{2\sqrt x }} - \cos x} \right) \cr
& y' = \sqrt {\frac{y}{x}} - \frac{{2\sqrt y \cos x}}{5} \cr
& {\text{Evaluate at the point }}\left( {4\pi ,\pi } \right) \cr
& y' = \sqrt {\frac{\pi }{{4\pi }}} - \frac{{2\sqrt \pi \cos 4\pi }}{5} \cr
& y' = \frac{1}{2} - \frac{{2\sqrt \pi }}{5} \cr
& y' = \frac{{5 - 4\sqrt \pi }}{{10}} \cr} $$
