Explanation

(b)
On taking log in the given equation, we get
log $y = \log{(1 - x) + \log{(2 - x) + ... + \log{(n - x)}}}$
$\ \ \frac{1}{y}\frac{dy}{dx} = \frac{1}{(1 - x)}( - 1) + \frac{1}{(2 - x)}( - 1) + ... + \frac{1}{(n - x)}( - 1)$
$\frac{dy}{dx} = y$
$\left\lbrack \frac{\begin{array}{r}
(2 - x)(3 - x)\ldots(n - x) + \\
(1 - x)(3 - x)\ldots(n - x) + ...
\end{array}}{y} \right\rbrack( - 1)$
$\Rightarrow$ $\left( \frac{dy}{dx} \right)_{x = 1} = 1.2\ldots..(n - 1)( - 1)$
$= ( - 1)(n - 1)!$