Complex Derivative: $\frac{d}{dz} (z+\overline{z} -z^3+z\overline{z})$

Question

How can I compute the following complex derivative

$$\frac{d}{dz} (z+\overline{z} -z^3+z\overline{z})$$

I know that $\frac{d}{dz} (z\overline{z})=z$ but not sure about the above.

Original problem:

1. $\displaystyle \frac{\partial}{\partial \bar{z}}(z+\bar{z}-z^3+z\bar{z})$.

2. $\displaystyle \frac{\partial}{\partial \bar{z}}((z+\bar{z}+16z^8-\bar{z})(-z^4+z^7))$

Answer 1

I am so confused. There is no "Complex Derivative" page in Wikipedia but I found it in Mathworld: "Complex Derivative".

Let's try to understand this: $\frac{d}{dz}(z\bar z)$. The function is $f(z)=x^2+y^2$ so, $u(x,y)=x^2+y^2$ and $v(x,y)=0.$ Then $u_x=2x\neq v_y=0$ and $u_y=2y\neq-v_x=0$. So, $f(z)$ is not complex differentiable and has no complex derivative, except at the origin.

Similarly, $\frac{d}{dz}(z+\bar z-z^3+z\bar z)$ does not exist on any open region of the complex plane.