Given the attempted derivative at http://www.wolframalpha.com/input/?i=d%28a%28x%29+choose+b%28x%29%29%2Fdx, I believe the result from Wolfram Alpha is off by a factor of 2 or, at least I get a result equal to twice as much WA does when I distribute du/dx and dv/dx thru their respective terms. Where did I go wrong?
Note: The u and v come into play when using the chain rule, as seen in the step-by-step solution. Also, a and b are continuous functions.
Well, here are WolframAlpha's steps:
Ultimately, it's just a matter of applying the multi-variate chain rule (shown in the WA output) together with the fact that the partial derivatives of the binomial can be expressed in terms of the digamma function. Specifically:
$$\frac{\partial}{\partial x} \binom{x}{y}=\binom{x}{y} (\psi ^{(0)}(x+1)-\psi^{(0)}(x-y+1))$$
and
$$\frac{\partial}{\partial y}\binom{x}{y}=\binom{x}{y} (\psi ^{(0)}(x-y+1)-\psi^{(0)}(y+1)).$$