# (x!)'

#### TruthSeeker

##### Fancy Virtual Reality Monkey
Valued Senior Member
Good luck!

Does anybody know the derivative of x!?
Prove it!

Let me try...

lim [(x+h)! - x!] / h
h->0
lim [(x+h)(x+h-1)(x+h-2)....x! - x!] / h
h->0
lim [(x+h)(x+h-n)(x+h-n-1)....x! - x!] / h
h->0
...
This is tough...!
...
I honestly don't know how to continue...

I think we would need some restrictions...

In order to have any chance of a derivative existing at a point, your function must be defined on an open interval containing that point. The factorial function x! is only defined when x is a non-negative integer, so you cannot differentiate it.

However, there is good news. The factorial function can be extended to (nearly) all reals by means of the Gamma function, G(x). This is a real valued function defined in terms of an integral and we have G(n+1)=n! everywhere n! is defined. This Gamma function can be differentiated easily enough.*

I'm short on time so I'm going to cop-out and point to this link for more info where they differentiate Gamma using logarithmic differentiation:

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

*note that Gamma is usually denoted with the capital greek letter gamma (as in the link) and it can actually be extended to the entire complex plane (except poles at non-positive integers)

Cool!!! Although mind-bending!
Too bad someone already worked on it. I was excited to think about it in my math class...

What about... (nPr)' and (nCr)' ?

Since those are just more factorials I'd wager it would follow the same conditions as above.

The Gamma function has been beaten severely to death by mathematicians over the years. It turns up all over the place, so it was pretty naturaly for it to be studied extensively. It's a pretty basic function actually, similar to the trig ones. In fact:

G(x-1)G(x)=Pi/sin(Pi*x)

This might seem like a mind-bending equation, but if you've studied some complex analysis and you think about the poles the functions on both side of the equation have, it's less mysterious.

nCr and nPr are defined in terms of factorials, so you could just think of them as products and quotients of Gamma. One thing to keep in mind though, is you have two variables n and r, so things get more complicated unless you restrict one of them to be a constant.

shmoe said:
The Gamma function has been beaten severely to death by mathematicians over the years. It turns up all over the place, so it was pretty naturaly for it to be studied extensively. It's a pretty basic function actually, similar to the trig ones. In fact:

G(x-1)G(x)=Pi/sin(Pi*x)
How does that connect to "!"?

This might seem like a mind-bending equation, but if you've studied some complex analysis and you think about the poles the functions on both side of the equation have, it's less mysterious.
*stares at equation...................*
*shrugs...

nCr and nPr are defined in terms of factorials, so you could just think of them as products and quotients of Gamma. One thing to keep in mind though, is you have two variables n and r, so things get more complicated unless you restrict one of them to be a constant.
Yes, that's why I asked it. Partial differentiation! Yuhoooo!!!

TruthSeeker said:
How does that connect to "!"?

Gamma is quite obviously related to factorial. Gamma is related to sine by that formula. So your friendly neighbourhood factorial is some related to sine. It doesn't seem like there should be any connection between Gamma (which remember, is an extention of factorial) and trig functions, but it's actually quite natural.

Hey Truth Seeker, this ties in very well with your other question about imaginary numbers. Once you expand the range of numbers into a two-dimensional domain, a whole lot of stuff starts to come together. To think that factorials, which we originally thought could only apply to integers, have a relation to trigonometric functions, is pretty amazing.

All made possible by the power of i.

One key to this will be for you to figure out what the value of e^iPi is.

It's a shame that trig was my worst math subject. It seems to be involved with almost every aspect of the universe.

Gamma(n+1) = n!

That's all cool, but why do we have to wait until the 2nd or 3rd year of university to learn that!? It's kinda dumb. This semester, I'm taking "Finite Mathematics" and I'm learning that Venn diagram stuff and probability stuff all over again. It is at least the fith time I've learnt that!!! Like... gimme a break! Why do I have to see that once again. It makes me sick, really. I wish I didn't need to waste my time with that anymore. The worse is that I don't feel motivated to study it again, so my marks just drop... :/

shmoe said:
Gamma is quite obviously related to factorial. Gamma is related to sine by that formula. So your friendly neighbourhood factorial is some related to sine. It doesn't seem like there should be any connection between Gamma (which remember, is an extention of factorial) and trig functions, but it's actually quite natural.
That's really bizarre, cause all values of sine are between -1 and 1... :/
Well, ok... maybe I'm thinking about the y axis... But it is still weird...

Fraggle Rocker said:
One key to this will be for you to figure out what the value of e^iPi is.
Why? I tried to do that in my calculator, but I don't have an "i" button....

It's a shame that trig was my worst math subject. It seems to be involved with almost every aspect of the universe.
It's one of my best...
Yuhooooo!

TruthSeeker said:
That's all cool, but why do we have to wait until the 2nd or 3rd year of university to learn that!? It's kinda dumb. This semester, I'm taking "Finite Mathematics" and I'm learning that Venn diagram stuff and probability stuff all over again. It is at least the fith time I've learnt that!!! Like... gimme a break! Why do I have to see that once again. It makes me sick, really. I wish I didn't need to waste my time with that anymore. The worse is that I don't feel motivated to study it again, so my marks just drop... :/

If it's your 5th or 6th time taking it, I have to wonder how your marks are dropping. Shouldn't it be routine by now?

TruthSeeker said:
That's really bizarre, cause all values of sine are between -1 and 1... :/
Well, ok... maybe I'm thinking about the y axis... But it is still weird..

It does look pretty strange. One thing to keep in mind is in the equation I gave relating Gamma to sine, the sine function was in the denominator, so the right hand side wanders off to infinity near the zeros of sine. The relations between all these functions, trig functions, hyperbolic trig functions, Gamma, e<sup>x</sup>, etc. will make more sense once you've studied some complex analysis (which you can do on your own).

shmoe said:
If it's your 5th or 6th time taking it, I have to wonder how your marks are dropping. Shouldn't it be routine by now?
Have you ever studied economics? In economics, there's something called marginal utility, which is the amount of satisfaction that a customer gets from a product after using it (or buying it) once again. This marginal utility drops as the quantity increases, because after consuming too much of one thing, you simply get tired of it. The same thing applies for Math. I've done it so many times that it became completely repetitive and useless for me. With less motivation to do the stuff, my marks finish dropping.

It does look pretty strange. One thing to keep in mind is in the equation I gave relating Gamma to sine, the sine function was in the denominator, so the right hand side wanders off to infinity near the zeros of sine. The relations between all these functions, trig functions, hyperbolic trig functions, Gamma, e<sup>x</sup>, etc. will make more sense once you've studied some complex analysis (which you can do on your own).
Isn't it hard to study on my own? How can I do it?

TruthSeeker said:
Isn't it hard to study on my own? How can I do it?

Not that hard.
Step 1- pick up textbook and open to section 1