Answer
$${\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{6}}} = - 3$$
Work Step by Step
$$\eqalign{
& y = \csc 3x + \cot 3x,{\text{ }}\left( {\frac{\pi }{6},1} \right) \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\csc 3x + \cot 3x} \right] \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\csc 3x} \right] + \frac{d}{{dx}}\left[ {\cot 3x} \right] \cr
& {\text{Recall that }}\frac{d}{{dx}}\left[ {\csc u} \right] = - \csc u\cot u \cdot u', \cr
& \frac{d}{{dx}}\left[ {\cot u} \right] = - {\csc ^2}u\frac{{du}}{{dx}}{\text{ }} \cr
& {\text{Then,}} \cr
& \frac{{dy}}{{dx}} = - \left( {\csc 3x\cot 3x} \right)\frac{d}{{dx}}\left[ {3x} \right] + \left( { - {{\csc }^2}3x} \right)\frac{d}{{dx}}\left[ {3x} \right] \cr
& \frac{{dy}}{{dx}} = - 3\csc 3x\cot 3x - 3{\csc ^2}3x \cr
& {\text{Evaluating at }}\left( {\frac{\pi }{6},1} \right) \cr
& {\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{6}}} = - 3\csc 3\left( {\frac{\pi }{6}} \right)\cot 3\left( {\frac{\pi }{6}} \right) - 3{\csc ^2}3\left( {\frac{\pi }{6}} \right) \cr
& {\left. {\frac{{dy}}{{dx}}} \right|_{x = \frac{\pi }{6}}} = - 3 \cr} $$
