Theorem: The derivative of the natural log function, y=ln(x), is given by

d/dx ln(x) = 1/x

This is easily seen by implicit differentiation of e^y=x, which yields dy/dx e^y = 1, and therefore,

dy/dx = 1/e^y = 1/x

The chain rule yields

d/dx ln ( u(x) ) = 1/u(x) du/dx

It is often easier to evaluate ln(u(x)) prior to differentiating, in order to simplify.

Special Facts:

d/dx log_a(x) = 1/(x ln(a) )
d/dx a^x = a^x/ln(a)

Logarithmic Differentiation

1. Take logarithms of both sides of y=f(x)
2. Differentiate implicitly with respect to x
3. Solve for y'(x)

Last modified Mon Oct 21 15:31:45 CDT 1996