Mark two readings of a changing quantity, join them with a straight line, and that line is the secant. Its slope is the average rate of change over the interval between them — same number, three names, depending on who is in the room. In symbols, $m = \dfrac{y_2 - y_1}{x_2 - x_1}$: the rise $y_2 - y_1$ on top, the run $x_2 - x_1$ underneath, subscript 1 for the first reading and 2 for the second.

Two disciplines carry this lesson. First, subtract in the same order top and bottom — swap the points in the numerator only and the sign comes out backwards, which is how a filling tank ends up reported as a draining one. Second, a slope in an applied problem is never a bare number: its unit is the y-unit over the x-unit, so it reads L/min, $/kg, °C/h. And a negative slope is an ANSWER, not an error — it says the responding quantity is falling, and falling is a perfectly respectable thing for a quantity to do.

Hold on to what this quotient becomes. Slide $x_2$ toward $x_1$ until the interval nearly closes, and the value the ratio settles on is the derivative — the instantaneous rate. Everything in this chapter is that same quotient with the interval still open.