# Derivative of Identity Function/Real

## Theorem

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

Then:

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

## Proof

The identity mapping is defined as:

$\forall x \in \R: \map {I_\R} x = x$

Thus:

 $\displaystyle \map {I_\R'} x$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {\map {I_\R} {x + \delta x} - \map {I_\R} x} {\delta x}$ Definition of Differentiation $\displaystyle$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {\paren {x + \delta x} - x} {\delta x}$ Definition of Identity Mapping $\displaystyle$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {\delta x} {\delta x}$ $\displaystyle$ $=$ $\displaystyle 1$

$\blacksquare$