For-f-x-x-Is-there-a-defined-1-f-x-2-f-x-dx-

Question Number 2231 by Filup last updated on 10/Nov/15

For f (x) = x!

Is there a defined :

(1) f^{'} (x)

(2) \int f (x) d x

Commented by Filup last updated on 10/Nov/15

Is this a correct assumption for (1) :

y = x!

y = x (x - 1) (x - 2) \dots

∴ y = \underset{i = 1}{\prod^{x}} i

U s i n g C h a i n r u l e :

∴ y^{'} = [(x - 1) (x - 2) \dots] + [x (x - 2) \dots] + \dots

y^{'} = \frac{y}{x} + \frac{y}{x - 1} + \dots = y (\frac{1}{x} + \frac{1}{x - 1} + \dots + 1)

y^{'} = y \underset{i = 1}{\sum^{x - 1}} (\frac{1}{i})

∴ y^{'} = y \cdot H (x - 1)

where H (n) is the sum to the n th

harmonic term

y = \underset{i = 1}{\prod^{x}} i

y^{'} = y \underset{i = 1}{\sum^{x - 1}} \frac{1}{i}

Note : I have no knowledge of differentiating

this function, so this is only a guess attempt

Commented by 123456 last updated on 10/Nov/15

f (x) = x! = Γ (x + 1)

f^{'} (x) = ψ (x + 1) x!

ψ (x) = \frac{d}{d x} \ln Γ (x) = \frac{Γ^{'} (x)}{Γ (x)}

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