Differentiation of Functions of a Complex Variable 2: Differentiation Formulas

 If $c$ is a constant complex number, then
\begin{equation}
\frac{dc}{dz}=0
\end{equation}
If $n$ is a positive integer, then
\begin{equation}
\label{eq:powerrule}
\frac{dz^n}{dz}=nz^{n-1}
\end{equation}
This formula is called the power rule and it remains valid when $n$ is a negative integer provided $z\ne 0$.

If $c$ is a constant complex number, then
\begin{equation}
\frac{d[cf(z)]}{dz}=c\frac{df}{dz}
\end{equation}
\begin{align}
\frac{d}{dz}[f(z)+g(z)]&=\frac{df(z)}{dz}+\frac{dg(z)}{dz}\\
\frac{d}{dz}[f(z)g(z)]&=\frac{df(z)}{dz}g(z)+f(z)\frac{dg(z)}{dz}\\
\frac{d}{dz}\left[\frac{f(z)}{g(z)}\right]&=\frac{\frac{df(z)}{dz}g(z)-f(z)\frac{dg(z)}{dz}}{[g(z)]^2}
\end{align}
The first two formulas are the linearity, i.e. the complex differentiation $\frac{d}{dz}$ is linear, the third formula is the product rule or the Leibniz rule, and the fourth formula is the quotient rule.

If $W=g(w)$ and $w=f(z)$, then
\begin{equation}
\label{eq:chain}
\frac{dW}{dz}=\frac{dg}{dw}\frac{dw}{dz}
\end{equation}
This is the chain rule.

Example. $$\frac{d}{dz}(2z^2+i)^5=20z(2z^2+i)^4$$