Derivative of Identity Function/Complex

Theorem

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

Then:

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

Corollary

$\map {\dfrac {\d} {\d x} } {c x} = c$

where $c$ is a constant.

Proof

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

Thus:

\(\ds \map {I_\C'} z\)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\map {I_\C} {z + h} - \map {I_\C} z} h\)

Definition of Derivative of Complex Function

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\paren {z + h} - z} h\)

Definition of Identity Mapping

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac h h\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$