Let me try to explain more
Write F as (x-a) times G(x) where G is all the factors of F except the first one.
its derivative is (using the product rule)
F'(x) = (x-a) G'(x) + (x-a)'G(x)
so far so good.
Now for the value x = a, the first term vanishes, The derivative of
F(x) = (x-a)(x-b)(x-c)(x-d)(x-e)(x-f)
F'(a) = (a-b)(a-c)(a-d)(a-e)(a-f)
That is a very simple expression to remember, but not so easy to derive!