Exponential is Strictly Increasing

From ProofWiki
Jump to navigation Jump to search


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

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


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.


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.