Derivative of Identity Function

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Theorem

Let $X$ be either set of either the real numbers $\R$ or the complex numbers $\C$.

Let $I_X: X \to X$ be the identity function.


Then:

$\map {I_X'} x = 1$

where $\map {I_X'} x$ denotes the derivative of $I_X$ with respect to $x$.


This can be presented for each of $\R$ and $\C$:


Derivative of Identity Function for Real Numbers

Let $I_\R: \R \to \R$ be the identity mapping on the real numbers $\R$.


Then:

$\map {I_\R'} x = 1$


Derivative of Identity Function for Complex Numbers

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.


Also presented as

Note that this can be written:

$\map {D_x} x = 1$

Using Leibniz's notation for derivatives it can also be stated as:

$\dfrac {\d x} {\d x} = 1$