Derivative of Logarithm Function

Natural Logarithm
Let $\ln x$ be the natural logarithm function.

Then:
 * $D_x \left({\ln x}\right) = \dfrac 1 x$

General Logarithm
Let $\log_a x$ be the logarithm function to base $a$.

Then:
 * $D_x \left({\log_a x}\right) = \dfrac {\log_a e} x$

Proof for Natural Logarithm
Follows directly from the definition of the natural logarithm function as the primitive of the reciprocal function.

Proof for General Logarithm
From Change of Base of Logarithm, we have:
 * $\log_e x = \dfrac {\log_a x} {\log_a e}$

from which:
 * $\log_a x = \log_a e \; \log_e x$

In this context, $\log_a e$ is a constant.

The result then follows from the above result and Derivative of Constant Multiple.