Exponential and Logarithmic Differentiation

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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 a^x and is defined as ^x = e^{(ln a)x}.
If a = 1, then y = 1^x = 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 log_ax and is defined as log_ax = (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[a^x] = (lna)a^x
d/dx[a^u] = (ln a)a^u du/dx
d/dx[lob_ax] = 1/((ln a)x)
d/dx[log_au] = 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] = nx^n-1
d/dx[uⁿ] = nu^n-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