Chapter 4.5, Problem 56E

This exercise shows another way to derive the formula for the derivative of the natural logarithm function using the definition of the derivative.

a. Using the definition of the derivative, show that $\frac{d\left(\ln x\right)}{dx}=\underset{h\to 0}{\lim }\ln {\left(1+\frac{h}{x}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$h$}\right.}$.

b. Eliminate h from the result in part a using the substitution $h=x/m$ to show that $\frac{d\left(\ln x\right)}{dx}=\underset{m\to \infty }{\lim }\ln {\left[{\left(1+\frac{1}{m}\right)}^m\right]}^{1/x}$

c. What property should the function g have that would yield $\underset{m\to \infty }{\lim }g\left(h\left(m\right)\right)=g\left(\underset{m\to \infty }{\lim }h\left(m\right)\right)$? Assuming that the natural logarithm has this property, and using the result about ${\left(1+1/m\right)}^m$ from Section 2.1, show that $\frac{d\left(\ln x\right)}{dx}=\ln e^{1/x}=\frac{1}{x}$.