Definition of a Function

Notation. A variable y is said to be a function of another variable x if, when x is given, y is determined.

The symbols f( x), F( x), etc., represent various functions of x.

The symbol f( a) represents the value of f( x) when x = a.

Definition of a Derivative

Notation. Let y = f( x). If ? x is any increment (increase or decrease) given to x, and ? y is the corresponding increment in y, then the derivative of y with respect to x is the limit of the ratio of ? y to ? x as ? x approaches zero, that is,

Some Relations Among Derivatives

If x = f( y), then

If x = f( t) and y = F( t), then

If y = f( u) and u = F( x), then

Table of Derivatives

Functions of x are represented by u and v, and constants are represented by a, n, and e.

Slope of a Curve. Tangent and Normal (Fig. 3-1)

Figure 3-1

The slope of the curve (slope of the tangent line to the curve) whose equation is y =