Derivative of Identity Function/Complex

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Theorem

Let $I_\C: \C \to \C$ be the identity function.


Then:

$\map {I_\C'} z = 1$


Corollary

$\dfrac {\mathrm d} {\mathrm d x} \left({c x}\right) = c$

where $c$ is a constant.


Proof

The identity function is defined as $\forall x \in \C: \map {I_\C} z = z$.

Thus:

\(\displaystyle \map {I_\C'} z\) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map {I_\C} {z + h} - \map {I_\C} z} h\) Definition of Derivative of Complex Function
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\paren {z + h} - z} h\) Definition of Identity Mapping
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac h h\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$