It may be easier to differentiate, if you rewrite the log.

ln\left(\sqrt{\frac{x-5}{6x-9}}\right)=\frac{1}{2}\left[ln(x-5)-ln(6x-9)\right]

Now, it should be easier to find the derivative.

\frac{d}{dx}\left[\frac{1}{2}ln(x-5)\right]=\frac{1}{2(x-5)}

\frac{d}{dx}\left[ln(6x-9)\right]=\frac{d}{dx}[ln(3(2x-3))]=\frac{d}{dx}[ln(3)+ln(2x-3)]=\frac{1}{2x-3}

Now, put them together.

\frac{1}{2(x-5)}-\frac{1}{2x-3}=\frac{7}{2(x-5)(2x-3)}