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:


 * on the open interval $\left ({0 \,.\,.\, +\infty} \right)$


 * on the open interval $\left ({-\infty \,.\,.\, 0} \right)$

Proof
Let $x \in \left ({0 \,.\,.\, +\infty} \right)$.

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.

The proof for $x \in \left ({-\infty \,.\,.\, 0} \right)$ is similar.