It is well-known by any student of calculus that the derivative of the natural log function, ln(x), is one over x or 1/x. What is not so well-known is the proof of this fact. What follows is a step-by-step proof which shows every last step.
The first step involves the limit definition of the derivative. The second step utilizes the fact that the difference of two logarithm functions is the logarithm of the quotient. The third step rewrites division as multiplication by the reciprocal. The fourth step involves the basic algebra of writing a single fraction as two fractions. The fifth step uses a substitution and the exponent law of logarithms to get a limit that is equal to e as the limit variable approaches zero.
Note that the continuity of the functions involved allow us to move the limit operation inside the natural log function. Also observe that 1/x behaves like a constant with respect to the limit variables h and u which allows us to pull it outside of the limit operation.