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.

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