Question

Differentiate the function w.r.t. $x$.
$\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

Answer 1

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