Calculus
Derivative as a Limit
Two points on a curve give one honest slope. Slide the second point in and watch the secant become the tangent, then zoom in until the curve itself is a straight line.
secanttangent
Click a line to select it, scroll or pinch to zoom
Slope of the secant—
—
Function
Which line
as drawn×1000
Values
f(x)—
f′(x)—
secant slope—
tangent slope f′(x0)—
gap between them—
curve off the tangent, on screen—
What to observe
- Shrink h and the gap between the two slopes shrinks with it. The number the secant settles on is f′(x₀).
- Zoom in on a smooth curve and it straightens: by ×1000 it sits on the tangent to within a pixel, and its slope is f′(x₀). Locally, every smooth curve is a line.
- Zoom in on the corner of |sin x| at x₀ = 0 and nothing straightens: the V looks the same at every magnification. The slopes from the left and the right (−1.6 and +1.6) disagree, so f′(0) does not exist.
- On ½eˣ the tangent slope always equals the height f(x₀): the curve is its own derivative.
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