Exponential and Logarithmic Differentiation

Exponential Functions to Base a
If a is a positive real number (a does not = 1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined as x = e(ln a)x.
If a = 1, then y = 1x = 1 is a constant function.

Logarithmic Functions to Base a
If a is a positive real number (a does not = 1) and x is any real number, then the logarithmic function to the base a is denoted by logax and is defined as logax = (1/ln a) ln x

Derivatives for Bases other than e
Let a be a positive real number (a does not = 1) and let u be a differentiable function of x.
d/dx[ax] = (lna)ax

d/dx[au] = (ln a)au du/dx

d/dx[lobax] = 1/((ln a)x)

d/dx[logau] = 1/((ln a)u) du/dx

The Power Rule for Real Exponents
Let n be any real number and let be a differentiable function of x.
d/dx[xn] = nxn-1

d/dx[un] = nun-1 du/dx

Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x.
1. d/dx[ln x] = 1/x, x > 0

But don't forget the Chain rule!

2. d/dx[ln u] = u'/u, u > 0