Derivative of Binomial Coefficient $\binom{2N}{N-x}$ with respect to $x$

Question

I've got $\binom{2N}{N-x}$ and I'd like to take the derivative with respect to $x$. I know that I can take the derivative of $\binom{n}{k}$ w.r.t. n using logarithmic differentiation, but that's not going to work for this. I plug it into Mathematica and get $\binom{2N}{N-x}[\psi(1+N-x)-\psi(1+N+x)]$, where $\psi$ is the digamma function. Can anybody help me figure out how to get from point A to point B without blindly trusting Mathematica?

Answer 1

Recall that $${2N \choose N-x}=\frac{(2N)!}{(N-x)!(N+x)!}=\frac{(2N)!}{\Gamma(N-x+1)\Gamma(N+x+1)},$$ then take the logarithm of this formula, differentiate it with respect to $x$ and finally use the definition $\psi(x)=\left(\ln\Gamma(x)\right)'$ of the digamma function.