TABLE OF CONTENTS This appendix covers the definitions and properties of ordinary and partial derivatives, Taylor expansions in one and two variables, and some properties of the critical points (turning points) of functions.
We start by considering functions f( x) of one variable, and their derivatives.
Let I be an interval. A function f : I ? 
is differentiable at a point x ? I if the limit
exists, in which case the limit in (C.1) is the derivative of f at x, which we write as (d f/d x)( x) or f ?( x).
This basic definition implies the standard rules of differentiation. The product rule is
the quotient rule is
and the chain rule, which allows us to differentiate functions of functions, is [1]
Taylor s Theorem allows us to expand a function f as a power series about a point x 0 using its derivatives. Suppose that f has n + 1 derivatives, all of which are continuous functions, and that we use the notation
Then we can write
for some point y n ? ( x 0, x).
Provided that the remainder term
tends to zero as n tends to infinity, we can write f( x) as the power series
known as the Taylor expansion or Taylor series for f .
