Solution:

Let’s take $y=\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

$=\left(\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}\right)^{\frac{1}{2}}$

On both the sides taking log, we get

$\log y=\frac{1}{2}[\log (x-1)+\log (x-2)-\log (x-3)-\log (x-4)-\log (x-5)]$

$\frac{1}{y} \frac{d y}{d x}=\frac{1}{2}\left[\frac{1}{x-1} \frac{d}{d x}(x-1)+\frac{1}{x-2} \frac{d}{d x}(x-2)-\frac{1}{x-3} \frac{d}{d x}(x-3)-\frac{1}{x-4} \frac{d}{d x}(x-4)-\frac{1}{x-5} \frac{d}{d x}(x-5)\right]$

$\frac{d y}{d x}=\frac{1}{2} y\left[\frac{1}{x-1}+\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-4}-\frac{1}{x-5}\right]$

$\frac{d y}{d x}=\frac{1}{2} \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\left[\frac{1}{x-1}+\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-4}-\frac{1}{x-5}\right] \text { [using the value of y] }$