Finding the Derivative of Factorial Function

[coki2000]

Hi,
I want to find the derivative of factorial function f(x)=x! and i found this integral,

$$f(x)=x!=\int_{0}^{\infty}e^{-t}t^xdt$$ when i take derivative of this

$$\frac{d}{dx}f(x)=\frac{d}{dx}\int_{0}^{\infty}e^{-t}t^xdt=\int_{0}^{\infty}e^{-t}t^xlnxdx$$

How do i find this integral? Please help me. Thanks.

 

[arildno]

Incorrect differentiation!

You should get:
$$\frac{df}{dx}=\int_{0}^{\infty}\ln(t)e^{-t}t^{x}dt$$

There is no nice expression for the anti-derivative with respect to "t" here, so it must be evaluated numerically.

 

[coki2000]

Incorrect differentiation!

You should get:
$$\frac{df}{dx}=\int_{0}^{\infty}\ln(t)e^{-t}t^{x}dt$$

There is no nice expression for the anti-derivative with respect to "t" here, so it must be evaluated numerically.

Okey if i want to find

$$f'(2)=\int_{0}^{\infty}\ln(t)e^{-t}t^{2}dt$$

can i integrate it?

 

[Borek]

can i integrate it?

Have you actually read arildno's post?

 

[Count Iblis]

http://mathworld.wolfram.com/DigammaFunction.html

 

[arildno]

Okey if i want to find

$$f'(2)=\int_{0}^{\infty}\ln(t)e^{-t}t^{2}dt$$

can i integrate it?

To anything simple?

Nope.

 

[Count Iblis]

Okey if i want to find

$$f'(2)=\int_{0}^{\infty}\ln(t)e^{-t}t^{2}dt$$

can i integrate it?

Yes, http://mathworld.wolfram.com/Euler-MascheroniConstant.html" .