📐 The Derivative as a Limit

About this simulation

The derivative of a function at a point is defined as the limit of the difference quotient: f'(a) = lim(h→0) [f(a+h) − f(a)] / h. Geometrically, the difference quotient is the slope of the secant line through the two points (a, f(a)) and (a+h, f(a+h)) on the curve. As h shrinks toward zero, the second point slides along the curve toward the first, and the secant line rotates and converges onto the tangent line — the line whose slope is the instantaneous rate of change at a.

How it works

• The curve f(x) is plotted in 3D along the x–y plane, raised slightly in z for depth.
• Two markers sit on the curve at x = a and x = a + h; a red secant line connects them.
• A blue tangent line is drawn at x = a using the exact derivative f'(a).
• As h → 0, the secant's slope (the difference quotient) approaches the tangent's slope — the limit that defines f'(a).

Controls

• Function f(x) — choose between a cubic, a sine wave, and a parabola.
• Fixed point a — moves the point of tangency along the curve.
• h — the horizontal offset to the secant's second point; drag it toward zero.
• Animate h → 0 — automatically shrinks h so you can watch the convergence.
• ↺ Reset view — restores the default camera. Drag to rotate, scroll to zoom.

Did you know?

This limiting process — turning an average rate of change (the secant slope) into an instantaneous rate of change (the tangent slope) — is the single idea that launched differential calculus, independently formalized by Newton and Leibniz in the 17th century.