Secant → Tangent
The derivative is the slope of a curve at a single point. But "slope at one point" sounds impossible — slope needs two points. Here's how calculus sneaks around that, and you can watch it happen. For Grade 12.
Two points, then squeeze
Pick a point P on the curve. Add a second point Q a distance h away. The line through them — the secant — has a slope you can compute:
Now slide Q toward P — shrink h toward zero. The secant pivots, and in the limit it rests on the curve as the tangent. Its slope is the derivative, written f ′(a).
Drag Q toward P 🟢🟠
Drag the green point P to choose where; drag the orange point Q — or the h slider — to shrink the gap. Or hit "Shrink h → 0" and watch the orange secant settle onto the green tangent.
What to watch
The secant is measurable
For any real gap h, the orange slope is just rise over run — ordinary Grade 9 stuff. The number in the readout is exact, no limits needed yet.
The tangent is the limit
As h → 0, the orange secant slope closes in on the green number, f ′(a). It never divides by zero — it just gets arbitrarily close. That "arbitrarily close" is the whole idea of a limit.
Try each function. On x², drag P around and watch the derivative equal 2a every time (at a = 3 the slope is 6). On sin x, the tangent is flat exactly at the peaks — where cos x = 0. You're reading a function's derivative straight off its shape.