I'm working on an expository talk using the text Geometric Group Theory by Drutu & Kapovich. On page 722 they give the formula for the directional derivative of $f$ in the direction $e^{i\alpha}$ to be $$\partial_{\alpha}f=\partial f + e^{-2i\alpha} \overline{\partial}f$$ Where $\partial f= \frac{\partial f}{\partial z}$ and $\overline{\partial}f= \frac{\partial f}{\partial \overline{z}}$. Here is my attempt at deriving this formula: \begin{align*} \partial_{\alpha} f&= \cos(\alpha) \partial_x f + \sin(\alpha)\partial_y f \\ &=\cos(\alpha) \left(\partial f + \overline{\partial} f \right)+\sin(\alpha)i\left(\partial f - \overline{\partial} f \right) \\ &= e^{i\alpha} \partial f + e^{-i\alpha}\overline{\partial} f \end{align*}
If someone could help me find my mistake or explain where the extra factor of $e^{-i\alpha}$ is coming from I would really appreciate it.