$\displaystyle 10\sin{\frac{2\pi x}{365}} + 25$
You need to use the chain rule on $\displaystyle \frac{2\pi x}{365}$
$\displaystyle f'(x) = 10cos(\frac{2\pi x}{365}) \times \frac{2\pi}{365} = \frac{20\pi }{365} cos(\frac{2\pi x}{365})$
$\displaystyle f'(150) = \frac{20\pi }{365} cos(\frac{2\pi \times 150}{365})$