# Derivative Visualizer

Canonical: https://manyspace.aporia.systems/tools/derivative
Updated: 2026-10-13

The derivative is a limit you can watch: shrink h and see the secant become the tangent.

## Try this

- Melt a secant. On the parabola at a = 1, step h from 1 down to 0.01. Watch the dashed secant swing onto the mint tangent while the error readout collapses toward zero.
- Hunt the flat spots. Switch to sine and ride the probe slowly from −4 to 4. Every crest and valley pins the exact slope at 0 while the crossings peak near ±1 — that rhythm is the cosine hiding inside the sine.
- Feel exponential honesty. On 2ˣ, compare the secant and exact slopes at a = 2. The slope is nearly as big as the value itself — exponentials grow in proportion to where they already are, which is why they explode.

## How to read a derivative

Pick two points on a curve, h apart, and draw the line through them: that secant has slope (f(a+h) − f(a)) / h — the average steepness over that stretch. Now shrink h. The two points collide, the secant settles onto one line, and that limit line — the tangent — has the instantaneous slope f'(a). The derivative was never a formula first; it is what the secant becomes.

The error readout is the microscope: at h = 1 the secant can be wildly off, but each shrink divides the error roughly by the same factor. On the parabola the exact slope is 2x, on sine it is cos x — ride the probe and check the tangent against your gut at peaks (flat, slope 0) and crossings (steepest).

## Related tools

- [Integral Explorer](https://manyspace.aporia.systems/tools/integral)
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle)
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane)
- [Probability Lab](https://manyspace.aporia.systems/tools/probability)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series)