Differentiate the function with respect to $x$: $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$
Let $y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$.
Taking the logarithm on both sides,we obtain:
$\log y = \log \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$
$\Rightarrow \log y = \frac{1}{2} \log \left[\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}\right]$
$\Rightarrow \log y = \frac{1}{2} [\log \{(x-1)(x-2)\} - \log \{(x-3)(x-4)(x-5)\}]$
$\Rightarrow \log y = \frac{1}{2} [\log (x-1) + \log (x-2) - \log (x-3) - \log (x-4) - \log (x-5)]$
Differentiating both sides with respect to $x$,we obtain:
$\frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right]$
$\Rightarrow \frac{dy}{dx} = \frac{y}{2} \left( \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right)$
$\therefore \frac{dy}{dx} = \frac{1}{2} \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \left[ \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right]$
Taking the logarithm on both sides,we obtain:
$\log y = \log \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$
$\Rightarrow \log y = \frac{1}{2} \log \left[\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}\right]$
$\Rightarrow \log y = \frac{1}{2} [\log \{(x-1)(x-2)\} - \log \{(x-3)(x-4)(x-5)\}]$
$\Rightarrow \log y = \frac{1}{2} [\log (x-1) + \log (x-2) - \log (x-3) - \log (x-4) - \log (x-5)]$
Differentiating both sides with respect to $x$,we obtain:
$\frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right]$
$\Rightarrow \frac{dy}{dx} = \frac{y}{2} \left( \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right)$
$\therefore \frac{dy}{dx} = \frac{1}{2} \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \left[ \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right]$
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