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Secant to tangent: the derivative

Shrink the gap between two points and zoom in until the curve is straight.

A curve with a point on it. A blue secant line passes through that point and a second point a gap h further along; its slope is the average rate of change. As the gap slider shrinks h toward zero, the secant turns into the dashed amber tangent line, whose slope is the derivative. A zoom slider magnifies around the point until the curve looks straight.

From two points to one: the tangent

Interactive

secant slope (average)5.3 m/s

tangent slope (derivative)10.2 m/s

difference4.9

Curve

Moment t1Gap h between the two points1Zoom in on the point×1

The math, with your numbers

\frac{f(x+h) - f(x)}{h} = \frac{20.4 - 15.1}{1} = 5.3 \;\xrightarrow{\ h \to 0\ }\; 10.2

h(t) = 20t - 4.9t^2 \quad\Rightarrow\quad h'(t) = 20 - 9.8t

For the ball, the derivative of height is the velocity: positive going up, zero at the top, negative coming down.

Try this

→Shrink h. Watch the secant slope close in on the tangent slope.
→Ball: find the moment the tangent is flat. What is happening to the ball?
→Zoom ×50 on any point. What does the curve look like up close?

Want to know why it behaves like this?

The lesson “Rates of Change: Derivatives” explains it step by step: How can a car have a speed at a single instant?

Read the lesson