# Exponential is Strictly Increasing

## Theorem

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

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

Then:

The function $\map f x = \exp x$ 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.

$\blacksquare$

## Proof 2

For all $x \in \R$:

 $\displaystyle D_x \exp x$ $=$ $\displaystyle \exp x$ Derivative of Exponential Function $\displaystyle$ $>$ $\displaystyle 0$ Exponential of Real Number is Strictly Positive

Hence the result, from Derivative of Monotone Function.

$\blacksquare$