**Definition:**The rate of change [slope] of a function at a point is the limiting value of its*average*slope over an interval including that point, as the width of the interval shrinks to zero:

All the remaining Laws and Rules can be proven by algebraic manipulation of this definition.**Power Law:**

valid for*all*powers*p*, whether positive, negative, integer, rational, irrational, real, imaginary or complex.**Product Law:**The derivative of the product of two functions is*not*the product of their derivatives! Instead,

**Constant times a Function:**The*Product Law*gives

where*a*is a constant (, not a function of*x*). This is often referred to as "pulling the constant factor outside the derivative."

**Function of a Function:**Suppose*y*is a function of*x*and*x*is in turn a function of*t*.

(also known as the**Chain Rule**)**Exponentials:**

where*k*is any constant.**Natural Logarithms:**

2004-02-13