Exponential is Strictly Increasing

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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 $x \in \R$:

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

Hence the result, from Derivative of Monotone Function.