Derivative of Identity Function

From ProofWiki

Jump to: navigation, search

Theorem

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

Then $\forall x \in \R: I_\R^{\prime} \left({x}\right) = 1$.

Note that this can be more compactly written $D_x \left({x}\right) = 1$.

$\dfrac{\mathrm{d}}{\mathrm{d}{x}} \left({c x}\right) = c$

The identity function is defined as $\forall x \in \R: I_\R \left({x}\right) = x$.

Thus:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle I_\R^{\prime} \left({x}\right)$	$=$	$\displaystyle \lim_{\delta x \to 0} \frac {I_\R \left({x + \delta x}\right) - I_\R \left({x}\right)} {\delta x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of differentiation
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \lim_{\delta x \to 0} \frac {\left({x + \delta x}\right) - x} {\delta x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \lim_{\delta x \to 0} \frac {\delta x} {\delta x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 1$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

Follows directly from the above, and Derivative of Constant Multiple.

Using Leibniz's notation for derivatives $\left (\dfrac{\mathrm dy}{\mathrm dx}\right )$ this theorem can be stated as

$\displaystyle \frac{\mathrm dx}{\mathrm dx} = 1$

Which is not to say that derivatives are fractions, but the theorem is quite elegant this way.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle I_\R^{\prime} \left({x}\right)\)	\(=\)	\(\displaystyle \lim_{\delta x \to 0} \frac {I_\R \left({x + \delta x}\right) - I_\R \left({x}\right)} {\delta x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of differentiation
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{\delta x \to 0} \frac {\left({x + \delta x}\right) - x} {\delta x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{\delta x \to 0} \frac {\delta x} {\delta x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)