**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.**Operator Notation:**The symbol (read "derivative with respect to*x*") can be thought of as a mathematical "verb" (called an*operator*) which "operates on" whatever we place to its right. Thus

**Power Law:**The simplest class of derivatives are those of power-law functions:

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 (*i.e.*, 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*.

**Exponentials:**

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

**Antiderivatives:**We can "solve"*integrals*or "antiderivatives" the same way we "solve" long division problems: by trial and error guessing! Suppose we are given an explicit function*f*(*x*) [for example,*f*(*x*) =*x*^{2}] and told that*f*(*x*)*is the derivative of*a function*y*(*x*) which we would like to know - that is,

Well, we know that

so we must divide by 3 to get

where the constant term*y*_{0}(the value of*y*when*x*=0) cannot be determined from the information given - the derivative of any constant is zero, so such an*integral*is always undetermined to within such a*constant of integration*.

2004-02-13