I am seeking $\frac{\text{d}}{\text{d}z}\left(z\bar{z}\right)$ where $f(z)=z\bar{z}.$
And I know that I need to use the following definition of the derivative: $f'(z)=\lim_{\Delta z\to 0}{\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}}.$ However, I'm not sure if I'm using the definition correctly when I plug in $f(z)$: \begin{align*} f'(z)&=\lim_{\Delta z\to 0}{\frac{(z+\Delta z)(\overline{z+\Delta z})-z\bar{z}}{\Delta z}}\\&=\lim_{\Delta z\to 0}{\frac{\overline{\Delta z}(z+\Delta z)+\bar{z}\Delta z}{\Delta z}}\\&=\lim_{\Delta z\to 0}{\frac{\overline{\Delta z}(z+\Delta z)}{\Delta z}}+\lim_{\Delta z\to 0}{\frac{\bar{z}\Delta z}{\Delta z}}\\&=\lim_{\Delta z\to 0}{\frac{\overline{\Delta z}(z+\Delta z)}{\Delta z}}+\bar{z} \end{align*} Assuming that I've maneuvered the limit above properly, I'm not sure how to continue from the final line...