Calculus: what is the derivative?

Tutoring calculus, derivatives are often visited. Yet, what is the derivative?

In calculus, the derivative means the slope of a line tangent to the curve at that point. In other words, it means the direction the graph would follow from that point if, from there, it stopped curving and continued on its way without changing direction.

A way to picture the derivative might be to imagine a circle as a graph. The derivative at its top point would be horizontal, likewise at its bottom point. At its extreme left point, the derivative would be vertical. One can imagine following the circle between its left and top points to one where its derivative would be a line running at 45 degrees.

In business, with a graph modeling a trend, the derivative might model the “if this rate continues” scenario.

Source:

Larson, R.E., Hostetler, R.P. (1989). Calculus, part one, third edition. D.C. Heath and Company.