Reciprocal Function is Strictly Decreasing/Proof 1

Theorem
The reciprocal function:


 * $\operatorname{recip}:\R \setminus \left\{ {0} \right\} \to \R$, $x \mapsto \dfrac 1 x$

is strictly decreasing.

Proof
By the definition of negative powers:


 * $\dfrac 1 x = x^{-1}$

From the Power Rule for Derivatives:

As even powers are positive, $-x^{-2} < 0$ for all $x$ considered.

Thus from Derivative of Monotone Function, $\operatorname{recip}$ is strictly decreasing.