Reciprocal Function is Strictly Decreasing/Proof 1

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

By the definition of negative powers:


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

From Power Rule for Derivatives:

From Square of Real Number is Non-Negative:
 * $-x^{-2} < 0$

for all $x$ within the domain.

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

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