The derivative as the limit of secant slopes

An animation tracks secant lines through the fixed point (2, f(2)) on f(x) = x^2/4 + 1/2 as the second point slides in with h shrinking from 2 to 0.05. A live slope readout shows the secant slope approaching 1, and the secant line visibly rotates into the tangent line. It builds intuition for the derivative as a limit of difference quotients, useful for introducing f'(a) in an early calculus unit.

16:9 · every frame verified for overlaps, spacing and edges before rendering

The prompt that made it

Secant lines through (2, f(2)) on f(x) = x^2/4 + 1/2 with h shrinking from 2 to 0.05; the slope readout converges to 1 and the secant becomes the tangent, illustrating f'(a) = lim (f(a+h) - f(a))/h.

Make your own version

Make the next one in this series

• Repeat the same construction at a different point a to show the slope readout converging to a new value
• Overlay the graph of f'(x) being traced out as a moves across the curve
• Compare left-hand and right-hand secant limits to show why the derivative requires both to agree

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