Factorial Differentiation: How to differentiate x!

Differentiation of factorial functions. Differentiate x factorial. Differentiate x! If y = x! find dy/dx.

[dr.rahulgupta]

Sir, I understand, the differential of ln x is 1/ x, diff of ln ( x --1) is 1/ ( x--1), but how diff of ln 2 is 1/ 2 or diff of ln 3 is 1/ 3 or diff of ln 4 is 1/4 ? ( We can differentiate the RHS in reverse order also ) . In my opinion your method is WRONG . On the other hand, you can write x! as gamma of ( x+ 1) and then then differentiate the Gamma function !Gamma function is differentiable but factorial function is NOT .

[kazagucci]

This is a very naive and incorrect 'proof'. Firstly, the factorial function is only defined for the natural numbers and you obviously can't take the derivative of a discrete function.
second, your use of ellipses to denote repeated multiplication, x(x-1)...(2)(1), is very vague and not well defined and, as another comment pointed out, The derivative of ln(2) is not 1/2 so this notation is very problematic. To see why we can't differentiate this like a normal function, write x! using Pi notation, prod[ k=1, x ]( k ), and notice how the variable is in the bounds of the product for which we don't have a formula to differentiate.

The proper way to differentiate it would be using the Gamma function which would give you d/dx(Gamma(x+1)) = Gamma(x+1)*digamma(x+1) where the digamma function is the harmonic series minus the Euler-Mascheroni constant, which is missing from your answer, therefore it is incorrect.

[Anonymous-tqiu]

Incorrect. You have to use the gamma function to solve this problem