First of all, the equation of the difference quotient is:
It is used to find the slopes of the graph. To derive it, look at the picture below. The first intercepted point is (x, f(x)) and the second is (x+h, f(x+h)). On a side note, the red line is called a secant because it's intercepted at two points and a tangent line is when it intercepts only at one point. Using the two points, plug them into the slope formula: (Y2-Y1)/(X2-X1). Once you have substituted these points, you'll see that the x in the denominator can cancel, leaving an h in the denominator. Once done simplifying, you'll see that it looks exactly like the difference quotient. The difference quotient finds the slope of a tangent line to a graph. This is also known as the derivative.