Answer
$${\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{4}}} = 0$$
Work Step by Step
$$\eqalign{
& y = \frac{1}{2}\csc 2x,{\text{ }}\left( {\frac{\pi }{4},\frac{1}{2}} \right) \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\frac{1}{2}\csc 2x} \right] \cr
& {\text{Pull out the constant}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{2}\frac{d}{{dx}}\left[ {\csc 2x} \right] \cr
& {\text{Recall that }}\frac{d}{{dx}}\left[ {\csc u} \right] = - \csc u\cot u\frac{{du}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{2}\left( { - \csc 2x} \right)\left( {\cot 2x} \right)\frac{d}{{dx}}\left[ {2x} \right] \cr
& \frac{{dy}}{{dx}} = \frac{1}{2}\left( { - \csc 2x} \right)\left( {\cot 2x} \right)\left( 2 \right) \cr
& \frac{{dy}}{{dx}} = - \csc 2x\cot 2x \cr
& {\text{Evaluating at }}\left( {\frac{\pi }{4},\frac{1}{2}} \right) \cr
& {\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{4}}} = - \csc 2\left( {\frac{\pi }{4}} \right)\cot 2\left( {\frac{\pi }{4}} \right) \cr
& {\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{4}}} = 0 \cr} $$
