Question

Question #3fc87

Answer 1

`F'(a) = (a-b)(a-c)(a-d)(a-e)(a-f)`

Explanation:

`F' = (x-a) [(x-b)(x-c)(x-d)(x-e)(x-f)]' + (x-b)(x-c)(x-d)(x-e)(x-f)`
Evaluated at `x=a` the first term vanishes. The second is the answer

Answer 2

Let me try to explain more

Explanation:

Write F as (x-a) times G(x) where G is all the factors of F except the first one.

`F(x) = (x-a) G(x)`

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 `(x-a)` is `1` and the last term of the total derivative is G(a) but G(a) is nothing but the original F divided by x-a and with all x replaced with a.

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!