BQ #7: Unit V - Derivatives and the Area Problem

In order to find the difference quotient, we must work with secant lines. Now, what is a secant line? A secant line basically intercepts a parabola at TWO different points. In order to find the difference quotient, however, we need a tangent line. We get this tangent line by reducing the space between the two points of the secant line until the practically overlap. In the picture below, we see that "x" is the space between the origin and the first point of intersection between the parabola and the secant line. The space between the first and second points of intersection is called "h". This makes the second point "x+h". 

In Order to find the difference quotient, we use the equation m= (y2 - y1) / (x2-x1). Our values (as shown in the graph are;

y2 = f(x+h)

y1 = f(x)

x2 = x+h

x1 = x

With this in mind, we have a denominator of "x+h-x", which means that we cancel the "x" and are left with "h" in the denominator. We have a denominator of "f(x+h) -f(x)", which leaves us with the difference quotient of "f(x+h) -f(x) / h".