Answer
$$\frac{{\cos \left( {\ln x} \right)}}{x}$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\left( {\sin \left( {\ln x} \right)} \right) \cr
& {\text{use the rule }}\frac{d}{{dx}}\left( {\sin u} \right) = \cos u\frac{{du}}{{dx}},{\text{ letting }}u = \ln x \cr
& = \cos \left( {\ln x} \right)\frac{d}{{dx}}\left( {\ln x} \right) \cr
& = \cos \left( {\ln x} \right)\left( {\frac{1}{x}} \right) \cr
& {\text{multiply}} \cr
& = \frac{{\cos \left( {\ln x} \right)}}{x} \cr} $$