How is $\frac{du}{dz}=\frac{1}{2}\left(\frac{du}{dx}-i\frac{du}{dy}\right)$ and $\frac{du}{d\bar{z}}=\frac{1}{2}(\frac{du}{dx} + i\frac{du}{dy})$
How does the derivative with respect to the complex conjugate even make sense?