Unsure if what I am doing with this function is correct;

I need to differentiate the following function,

$\displaystyle f(x) = \log_{e^x}(\sin x)$

I have used the change of base property

$\displaystyle \log_{b}x = \frac{\log_{a}x}{\log_{a}b}$

Thereofre;

$\displaystyle f(x)=\frac {\ln(\sin x)} {\ln e^x} = \frac {\ln(\sin x)} {x}$