How is the difference quotient derived?
With cheery songs and an able memory, one can easily memorize the difference quotient formula. However, knowing how this formula is derived can reveal so much about functions and how they, along with their derivatives, work. The difference quotient formula is derived from the slope of a secant line that touches the function at two points.
Finding the slope of a secant line, a line that touches a function at two points, can be easy and applicable to any function if we use variables to represent points and distances. In the function depicted below, the pink secant line traverses the blue function at two points. The first point has a distance of x, and its y value can be denoted as f(x). The second point touched by the secant line can be written as the distances of x and h combined. This is because the distance is measured at 0 and includes x along with its measurement. Its y value can be denoted as f(x+h).
Having two points of a line is all that is required to use the slope formula and find the slope of the secant line.
The "any point in a function" refers to the distance between point 1 and point 2. That distance is also referred to as change in h (or deltah). As that distance gets closer and closer, nearing zero, it can be called a tangent line. When the difference quotient is evaluated at zero, it becomes the derivative.
