Exponential is Strictly Increasing

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Theorem

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

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


Then:

The function $f \paren x = \exp x$ 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 $x \in \R$:

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


Hence the result, from Derivative of Monotone Function.

$\blacksquare$


Sources