\begin{definition}[Definition:Derivative of Smooth Path/Complex Plane]
Let $\gamma: \closedint a b \to \C$ be a '''smooth path in $\C$'''.
Define the real function $x : \closedint a b \to \R$ by:
:$\forall t \in \closedint a b: \map x t = \map \Re {\map \gamma t}$
Define the real function $y: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b: \map y t = \map \Im {\map \gamma t}$
where:
:$\map \Re {\map \gamma t}$ denotes the real part of the complex number $\map \gamma t$
:$\map \Im {\map \gamma t}$ denotes the imaginary part of $\map \gamma t$.
It follows from the definition of a  smooth path that both $x$ and $y$ are continuously differentiable.
Let $\map {x'} t$ and $\map {y'} t$ denote the derivative of $x$ and $y$ {{WRT|Differentiation}} $t$.
The '''derivative of $\gamma$''' is the continuous complex function $\gamma': \closedint a b \to \C$ defined by:
:$\forall t \in \closedint a b: \map {\gamma'} t = \map {x'} t + i \map {y'} t$
\end{definition}