# Calculus in a Nutshell

• 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) = x2] 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 y0 (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.

Jess H. Brewer
2004-02-13