# Derivative of x!

1. Feb 7, 2010

### 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

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

2. Feb 7, 2010

### 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.

3. Feb 7, 2010

### coki2000

Okey if i want to find

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

can i integrate it?

4. Feb 7, 2010

### Staff: Mentor

Have you actually read arildno's post?

5. Feb 7, 2010

6. Feb 7, 2010

### arildno

To anything simple?

Nope.

7. Feb 7, 2010

### Count Iblis

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

Last edited by a moderator: May 4, 2017