# 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 $\map f x = \exp x$ 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.

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 14.4$