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 $x \in \R$:
\(\ds D_x \exp x\) | \(=\) | \(\ds \exp x\) | Derivative of Exponential Function | |||||||||||
\(\ds \) | \(>\) | \(\ds 0\) | Exponential of Real Number is Strictly Positive |
Hence the result, from Derivative of Monotone Function.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.4$