Exponential is Strictly Increasing

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:
 * The function $f \left({x}\right) = \exp x$ is strictly increasing.

Proof 1
By definition, the exponential function is the inverse of the natural logarithm function.

From Logarithm is Strictly Increasing, $\ln x$ is strictly increasing.

The result follows from Inverse of Strictly Monotone Function.

Proof 2
For all $x \in \R$:

Hence the result, from Derivative of Monotone Function.