Defining Average and Instantaneous Rate of Change at a Point

A curve doesn't have one slope the way a line does — it keeps turning. So to measure how fast f is changing between two points, you draw a secant line through them and compute its slope: [f(x) - f(p)] / (x - p). That's the average rate of change on the interval, a single number standing in for all the bending that happened in between. It tells you the net effect, not the moment-to-moment behavior.

To get the rate of change at a single point, slide the second point toward the first. As x creeps toward p, the secant line pivots and settles into the tangent line at p, and its slope settles into a limiting value — the instantaneous rate of change. This is exactly the limit definition of the derivative: f'(p) = lim as x→p of [f(x) - f(p)] / (x - p). The tangent line is not defined by touching the curve once or by hugging it without crossing; it's defined as whatever line the secant lines are approaching. That's why a line can cross straight through the curve at the point of tangency, which happens all the time at inflection points, and still be the true tangent.

This distinction sets up everything else in the unit. Average rate of change is just algebra — two points, one slope. Instantaneous rate of change needs a limit, which is why the derivative is built on limits before any shortcut rules show up. When the secant slopes approaching a point from the left and right don't agree on a common limit, as at a sharp corner, there's no tangent line and no derivative there — the function fails to be differentiable even though it's still perfectly continuous.