I obtained the following formula in Mathematica:

$$\frac{d}{dn}\ln\binom{n}{k} = H\_{n} - H\_{n-k}$$

where $H\_n$ are the harmonic numbers ($H\_n = \sum\_{i=1}^n 1/i$). But I have no idea how to prove it. Can someone help me? Or at least provide a reference to a textbook/paper?