Derivative of Reciprocal

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Theorem

Let $f: \R_{\ne 0} \to \R$ be the reciprocal function defined as:

$\map f x = \dfrac 1 x$


Then its derivative is given by:

$\map {f'} x = -\dfrac 1 {x^2}$


Proof

We have:

\(\ds \dfrac 1 x\) \(=\) \(\ds x^{-1}\)
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\dfrac 1 x}\) \(=\) \(\ds \paren {-1} x^{-2}\) Power Rule for Derivatives
\(\ds \) \(=\) \(\ds -\dfrac 1 {x^2}\)

$\blacksquare$


Sources