Question

Determine f'(0) f:r->r given by f(x)= x(x-1)(x-2).......(x-2019)

Answer 1

Answer:

f'(0) = - 2019!  (i.e. minus 2019 factorial).

Step-by-step explanation:

The product rule with many terms just says the derivative of u₁u₂u₃...uₙ is the sum of the terms obtained by differentiating one factor at a time; that is, it is

u'₁u₂u₃...uₙ + u₁u'₂u₃...uₙ + u₁u₂u'₃...uₙ + ... + u₁u₂u₃...u'ₙ

So here, we have

f'(x) = (x-1)(x-2)...(x-2019) + [ a whole heap of other terms that all start with the factor x ]

Putting x = 0, all the terms with x as a factor will vanish, so all that remains is the first term.  Thus

f'(0) = (-1)(-2)...(-2019) = - 2019!  (i.e. minus 2019 factorial).