BQ #7: Unit V: Derivation of the Difference Quotient

The difference quotient is derived through the slope of a secant line. We can use this image as a guide to figure out how:

The distance from the origin to Point A on the x-axis is "x". The distance from point A to point B on the x-axis is "h", so therefore the distance from the origin to Point B is "x+h". The distance on the y-axis from the origin to Point A is labeled as "f", and because the x-coordinate of Point A is "x", the y-coordinate of Point A is "f(x)", therefore the coordinate of Point A is: "(x, f(x))". likewise, the distance on the y-axis from Point A to Point B is "h", so the y-coordinate for point B is: "f(x+h)". Therefore, the coordinate for Point B is: "(x+h, f(x+h))".

The distance (slope) from Point A to Point B is also known as the secant line. We plug in the x and y coordinates of both points into the slope formula. Once we plug it in, we can simplify by canceling out x in the denominator. This gives us the difference quotient.