Is there a new way to calculate derivatives of factorials?

[dimension10]

I think I have found a formula for finding Successive derivatives of factorials, though it may have been found already. I have attached it to this post.

 

[chiro]

I think I have found a formula for finding Successive derivatives of factorials, though it may have been found already. I have attached it to this post.

Are you aware of Euler's Gamma Function?

http://en.wikipedia.org/wiki/Gamma_function

You can use that, along with the fact that Gamma(x) = (x-1)! for whole numbers x, along with the fundamental theorem of calculus to show what the derivative is for Gamma(x) not only for valid integers, but also for any valid real number as well.

 

[dimension10]

Are you aware of Euler's Gamma Function?

http://en.wikipedia.org/wiki/Gamma_function

You can use that, along with the fact that Gamma(x) = (x-1)! for whole numbers x, along with the fundamental theorem of calculus to show what the derivative is for Gamma(x) not only for valid integers, but also for any valid real number as well.

Yes, I am aware of the Gamma function, but I never thought that it would have any implications in this.

 

[henry_m]

I think I have found a formula for finding Successive derivatives of factorials, though it may have been found already. I have attached it to this post.

I'm afraid the 'derivative of the factorial function' doesn't exist. The function is only defined for non negative integer values so there is no meaningful concept of the slope of the function.

We could extend the function to include all positive real numbers, and the gamma function is a very natural way of doing this (though by no means unique). Then we can talk meaningfully about derivatives, but it's no longer the factorial function we're talking about.

 

[chiro]

I'm afraid the 'derivative of the factorial function' doesn't exist. The function is only defined for non negative integer values so there is no meaningful concept of the slope of the function.

We could extend the function to include all positive real numbers, and the gamma function is a very natural way of doing this (though by no means unique). Then we can talk meaningfully about derivatives, but it's no longer the factorial function we're talking about.

Henry_m is spot on. I should have mentioned that. Also are you aware of the fundamental theorem of calculus?

 

[3.1415926535]

It depends on how you define the factorial
If you define it like this x!=\prod_{n=1}^{x}n=1\cdot 2\cdot 3\cdot ...\cdot (x-1)\cdot x then the function f(x)=x! is only defined for the natural numbers. Therefore the graphical represantion of the function will be a list of unconnected points, so a derivative won't make sense

If you define it like this x!=\Gamma (x+1)=\int_{0}^{+\infty}t^{x}e^{-t}dt\forall x\geq0 then \frac{d}{dx}x!=\frac{d}{dx}\Gamma(x+1)=\int_{0}^{+\infty}\frac{d}{dx}t^{x}e^{-t}dt=\int_{0}^{+\infty}t^{x}e^{-t}\ln{t}dt

 

[pwsnafu]

I'm afraid the 'derivative of the factorial function' doesn't exist. The function is only defined for non negative integer values so there is no meaningful concept of the slope of the function.

Sorry but this is false. The domain of the factorial is a closed subset of the reals, and so http://en.wikipedia.org/wiki/Time_scale_calculus" can be used. In other words, there is a canonical way of defining a "derivative", which in this case is the forward difference operator. What is true is that the derivative from continuum analysis is not defined, but if we are talking about discrete analysis, there is no problem.

 

[HallsofIvy]

The website you link to talks about the "delta derivative" (also called the "Hilger derivative") NOT the standard derivative which is what is being discussed here.

 

[pwsnafu]

The website you link to talks about the "delta derivative" (also called the "Hilger derivative") NOT the standard derivative which is what is being discussed here.

Two things: if the domain of a function is the reals then the delta derivative is the standard derivative, that's the whole point of time scale calculus. But the reason I posted is that OP's PDF is calculating the delta derivative, albeit, not in a mathematically rigorous way.

PS. Reading my post back, it was probably too strong. Apologies to henry.