A difference quotient is a mathematical term that refers to a formula for finding the

secant line or

derivative of a

function. If we imagine the

graph of a

function and any two

points on that

graph, we can imagine them as being connected by a

line. This

line, which connects two

points on a

function, is called a

secant line. The

slope of the

secant line is equal to the

average rate of change of the

function over the

interval between the

points. As the two

points get closer together, we are closer to finding the

instantaneous rate of change as opposed to an

average rate of change. This is good because we are looking for a

derivative, an

expression for an

instantaneous rate of change. If we call the first point X, then its

x coordinate is (x). The second term is thus X, but further along the

function by a distance 'h'. The

average rate of change can thus be found by examining the value of (f(x) - f(x+h)) / h. The problem that immediately emerges when we are looking for the

slope at a

point, in effect the

slope of a

line tangent to that

point, is that when h has a value of

zero the above equation is

undefined. In order to overcome this, and glean a

derivative out of the

expression, we must find the

limit of (f(x) - f(x+h)) / h as h approaches

zero. By

simplifying this

expression one can find the

derivative. For example:

f(x) = x^2

Difference Quotient:

((x+h)^2 - (x)^2) / h

(x^2+2hx+h^2 - x^2) / h

(2hx + h^2) / h

(2x + h)
Find

Derivative Using

Limit:

lim (2x + h)

h->0

2x

The

derivative (f'(x)) of f(x) = x^2 is thus discovered to be 2x using a difference quotient.