Differentiating the binomial coefficient

A way to write the (usual) binomial coefficient is $$\binom{n}{r}= \frac{\prod_{i=0}^{r-1}(n-i)}{r!}.$$

In this expression $n$ does not have to be an integer for it to make sense. This is the (or at least one) way to extend the definition.

So $$\binom{x}{r}= \frac{\prod_{i=0}^{r-1}(x-i)}{r!}.$$

As small supplement:

A generalisation of the binomial coefficient is used in the binomial series representation \begin{align*} (1+x)^\alpha=\sum_{r=0}^\infty\binom{\alpha}{r}x^r\qquad\qquad |x|<1, \,\alpha\in\mathbb{C} \end{align*} where the binomial coefficient \begin{align*} \binom{\alpha}{r}=\frac{1}{r!}\alpha(\alpha-1)\cdots(\alpha-r+1) \end{align*} can be defined even for complex $\alpha$. This implies that we can consider the binomial coefficient as real-valued (or complex-valued) polynomial of degree $r$ \begin{align*} &f:\mathbb{R}\rightarrow\mathbb{R}\\ &f(x)=\binom{x}{r}\\ &\qquad=\frac{1}{r!}x(x-1)\cdots(x-r+1) \end{align*} which is so accessible to analytical operations (differentiation, etc.).

And you're right, it's not necessary to involve the Gamma-function here.