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:

\(\ds \map {I_\R'} x\)

\(=\)

\(\ds \lim_{\delta x \mathop \to 0} \frac {\map {I_\R} {x + \delta x} - \map {I_\R} x} {\delta x}\)

Definition of Derivative of Real Function

\(\ds \)

\(=\)

\(\ds \lim_{\delta x \mathop \to 0} \frac {\paren {x + \delta x} - x} {\delta x}\)

Definition of Identity Mapping

\(\ds \)

\(=\)

\(\ds \lim_{\delta x \mathop \to 0} \frac {\delta x} {\delta x}\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$