Derivative of Identity Function/Real

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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
\(\displaystyle \) \(=\) \(\displaystyle \lim_{\delta x \mathop \to 0} \frac {\delta x} {\delta x}\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$