Question

Differentiate the given functions w.r.t. x.

$\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

Answer 1

Taking log on both sides, we get

$logy=log{\left\{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}\right\}}^{1/2}=\frac{1}{2}\left[log\right(x-1\left)\right(x-2)-log(x-3\left)\right(x-4\left)\right(x-5\left)\right]=\frac{1}{2}\left\{log\right(x-1)+log(x-2)-log(x-3)-log(x-4)-log(x-5\left)\right\}$

Differentiating both sides w.r.t. x, we get

$\frac{1}{y}\frac{dy}{dx}=\frac{1}{2}\left[\frac{d}{dx}log\right(x-1)+\frac{d}{dx}log(x-2)-\frac{d}{dx}log(x-3)-\frac{d}{dx}log(x-4)-\frac{d}{dx}log(x-5\left)\right]=\frac{1}{2}\left\{\frac{1}{x-1}\right(1-0)+\frac{1}{x-2}(1-0)-\frac{1}{x-3}(1-0\left)\frac{1}{x-4}\right(1-0)-\frac{1}{x-5}(1-0\left)\right\}\Rightarrow \frac{dy}{dx}=\frac{y}{2}\{(\frac{1}{x-1}+\frac{1}{x-2})-(\frac{1}{x-3}+\frac{1}{x-4}+\frac{1}{x-5})\}=\frac{1}{2}\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}\{\frac{1}{x-1}+\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-4}-\frac{1}{x-5}\}$