Derivative of Identity Function

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:

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

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives