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.

$\blacksquare$