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.