Derivative of Identity Function/Real

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

$\blacksquare$