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Derivative from First Principle

The derivativ...

Question

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

0

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(- 1) (n - 1)!

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n! - 1

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{(- 1)}^{n - 1} (n - 1)!

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{(- 1)}^{n} (n - 1)!

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Solution

The correct option is B $(- 1) (n - 1)!$
$y = (1 - x) (2 - x) (3 - x) . . . (n - x)$
Differentiating both sides w.r.t. $x$ we have,
$\frac{d y}{d x} = [- (2 - x) (3 - x) (4 - x) . . . (n - x)] + [- (1 - x) (3 - x) (4 - x) . . . (n - x)] + . . . + [(1 - x) (2 - x) (3 - x) . . . (n - 1 - x)]$
At $x = 1 \frac{d y}{d x} = - 1 \times 2 \times 3 \times . . . \times (n - 1) = - 1 (n - 1)!$

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