Natural Logarithm Function is Differentiable

Theorem
The natural logarithm function is differentiable.

Proof 1
This proof assumes the definition of $\ln$ as :


 * $\ln x = \displaystyle \int_1^x \frac 1 t \ \mathrm dt$

As $\ln$ is defined as a definite integral, the result follows from the corollary to the first fundamental theorem of calculus.

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.