Is this differentiation of factorial valid?

Question

I have an expression like: $$\frac{d(n(x)!)^{-1}}{dx}$$ where $n(x)$ is given by $n(x)=floor(x \exp(-c))$ now can I do this like $-1(n(x))^{-2}(n(x)-1)(n(x)-1)! \frac{dn(x)}{dx}$ where i am using the method of finite differences and also some reference from this question. Is this the correct way?

Just one edit: Will it make a significant difference if we have partial derivative with respect to x rather then differentiation?

What? How did you derive the factorial function? Did you mean gamma function? — Ofek Gillon, May 05 '17 at 10:22
But the factorial is a discrete function, you cannot derive it — Ofek Gillon, May 05 '17 at 10:27
@Ankit The floor-function is not differentiable at integers. — Bobson Dugnutt, May 05 '17 at 10:35
Is there a way by which I can get approximate answer for this? I am actually doing differentiation wrt x, but the problem is this in the bigger function factorial comes, which unfortunately is a function of x. Now I do not know what to do, how should I solve the differentiation. — Userhanu, May 05 '17 at 10:37
note that the floor function between consecutive integer will be constant hence the first derivation is zero. Just look for intervals $n(x)!$ is constant. Also note that $$d(n(x)!)^{-1}/dx=(d(n(x)!)/dx) / (n(x)!)^2$$ if differentiable. — SaeidAli, May 05 '17 at 10:42

score 0 · Answer 1 · answered May 05 '17 at 10:43

0

The floor function is not differentiable, but you can try something with the gamma function $\Gamma$, which for integers satisfy

$$ \Gamma(n) = (n - 1)! $$ In this case, your then problem becomes finding the derivative

$$ \frac{{\rm d}}{{\rm d} x}(\Gamma(f(x) + 1))^{-1} $$

where

$$ f(x) = x e^{-c} $$

Hope it helps

answered May 05 '17 at 10:43

caverac

19,345

So you mean floor(x) in the interval $0<x<1$ is not differentiable which is not true. – SaeidAli May 05 '17 at 10:45
in the question for all integer $n$, if $ne^c<x<(n+1)e^c$ then the first derivative is zero. and for all points $x=ne^c$ is not differentiable. – SaeidAli May 05 '17 at 10:48

Is this differentiation of factorial valid?

1 Answers1