The natural logarithm

The natural logarithm function, $\ln$, is the inverse of the exponential function, $\exp$. Algebraically this means $\ln(\exp(a)) = a$ and $\exp(\ln(a)) = a$ so long as domain requirements are respected. To find the slope of the tangent line to $y = \ln(x)$ at the point $(a,\ln(a))$ we reason that if the slope of this line is m, then the slope of the tangent line to $y = \exp(x)$ at $(\ln(a),a)$ should be 1/m, since when the graph of $y = \exp(x)$ is reflected onto the graph of $y = \ln(x)$, the point $(\ln(a),a)$is carried onto the point $(a,\ln(a))$. Well, since $\exp'(x) = \exp(x)$,we know that the slope of the tangent line to $y = \exp(x)$ at $(\ln(a),a)$ is given by $\exp(\ln(a)) = a$!. Hence 1/m = a, that is m = 1/a is the slope of the tangent line to $y = \ln(x)$ at $(a,\ln(a))$. Hence we find that the rate of change of $\ln$ is given by $\ln'(a) = 1/a = a^{-1}$! Here is part of the missing piece of the power rule, insofar as here a > 0. It can be shown that if we define $f:(-\infty,\infty)\rightarrow(-\infty,\infty)$, $f(x) = \ln(\vert x\vert)$, then f'(x) = x^-1 for all $x\in(-\infty,0)\cup(0,\infty)$.