Natural Logarithm Function is Differentiable
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Theorem
The (real) natural logarithm function is differentiable.
Proof 1
This proof assumes the definition of $\ln$ as:
- $\ln x = \ds \int_1^x \frac 1 t \rd t$
As $\ln$ is defined as a definite integral, the result follows from the corollary to the first fundamental theorem of calculus.
$\blacksquare$
Proof 2
This proof assumes the definition of $\ln$ as the inverse of the exponential function.
As the Exponential Function is Differentiable, the result follows from the differentiability of inverse functions.
$\blacksquare$