# Derivative of Identity Function/Real

## Theorem

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

Then:

$I_\R' \left({x}\right) = 1$

## Proof

The identity function is defined as:

$\forall x \in \R: I_\R \left({x}\right) = x$

Thus:

 $\displaystyle I_\R' \left({x}\right)$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {I_\R \left({x + \delta x}\right) - I_\R \left({x}\right)} {\delta x}$ Definition of Differentiation $\displaystyle$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {\left({x + \delta x}\right) - x} {\delta x}$ Definition of Identity Function $\displaystyle$ $=$ $\displaystyle \lim_{\delta x \mathop \to 0} \frac {\delta x} {\delta x}$ $\displaystyle$ $=$ $\displaystyle 1$

$\blacksquare$