Slope

the derivative was just the direction you were facing.

touch the hillside -- a second point appears. drag the two together and the secant collapses into the tangent

A tangent line at a point is not a separate idea from a secant line through two points -- it is the limit of one. Touch anywhere on this hillside and a second point appears nearby; the line between them is only ever an estimate of the slope where you touched, and dragging the second point toward the first makes that estimate better. Close the gap enough and the two points become indistinguishable from a single point with a single direction -- the tangent, and its slope is the derivative. It settles exactly where the hillside is level, which is the whole reason a derivative is how you find a peak without ever seeing the whole hill at once.

Concepts: geometry, motion.

An interactive you play with one thumb, on Wundr.