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Derivative of a product (x - 1)(x - 2)...(x - n) at one of its roots

The same question, asked 2 times: 19 Apr 2025 S2*, 12 May 2026 S1. Every wording below, oldest first.

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  1. Question 1

    19 Apr 2025 S2

    106
    The derivative of y=(1−x)(2−x)y= (1 -x)(2 -x)(n-x) at x=1x= 1 is

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  2. Question 2

    12 May 2026 S1

    73
    If y=(x−1)(x−2)(x−3)⋯(x−100)y = (x - 1)(x - 2)(x - 3)\cdots(x - 100) and the value of dydx\dfrac{dy}{dx} at x=5x = 5 is equal to λ(100!100C5)\lambda\left(\dfrac{100!}{^{100}C_5}\right), then λ=\lambda = ...

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