1. Tardigrade
  2. Question
  3. Mathematics
  4. The derivative of y = (1 - x)(2 - x ) ldots (n - x) at x = 1 is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The derivative of $y = (1 - x)(2 - x )$ $\ldots (n - x)$ at $x = 1$ is equal to

Continuity and Differentiability

Solution:

$\because y = \left(1 - x\right)\left(2 - x\right)\ldots\ldots\left(n - x\right)$
Taking log on both sides, we get
$logy = log\left(1 - x\right) + log\left(2 - x\right) + ... + log\left(n - x\right)$
$\frac{1}{y} \frac{dy}{dx}=\frac{1}{\left(1-x\right)}\left(-1\right)+\frac{1}{\left(2-x\right)}(-1)+\ldots\ldots+\frac{1}{\left(n-x\right)}\left(-1\right)$
$\frac{dy}{dx}=-y\left[\frac{\left[\left(2-x\right)\left(3-x\right)\ldots\ldots\left(n-x\right)+\ldots\ldots\right]}{y}\right]$
$\therefore \left(\frac{dy}{dx}\right)_{x=1}=1\cdot2\ldots\ldots\left(n-1\right)\left(-1\right)$
$=\left(-1\right)\left(n-1\right)!$