Derivative Visualizer
The derivative is a limit you can watch: shrink h and see the secant become the tangent.
Tip: ride the probe to a peak — the tangent goes flat and the exact slope reads 0.
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).
Calculus makes sense once you've watched a limit happen
This visualizer is one small piece of how ManySpace teaches mathematics — every idea as something you can run, shrink, and push to the extreme.