Calculus
Derivative as a Limit
Two points on a curve give you one honest slope: rise over run. Slide the second point in and that slope keeps changing, but it changes towards something. Shrink h to nothing and the secant becomes the tangent, and its slope is the derivative.
Secant slope—
Tangent slope f′(x₀)—
Gap between them—
Click either line on the graph to select it. The chosen one lights up and its arithmetic is written out on the right.
What to observe
- The amber secant is the only slope you can measure honestly: two points, a rise and a run. The blue tangent touches at one point, so there is no run to divide by and no slope to measure directly.
- That is what the limit is for. Halve h and watch the gapreadout roughly halve too. The secant slope is not reaching the tangent by accident; it is closing in at a steady rate.
- Press Shrink h → 0. The triangle collapses, the two lines become one and the number on the right stops moving. Whatever it settles on is called f′(x₀), and nothing else about the curve was needed to find it.
- Pick ½eˣ. The tangent slope always equals the curve's own height f(x₀), which is the property that singles e out from every other base.
- Pick sin x and drag x₀ to a crest. The tangent goes flat, the secant does not, and the gap is at its largest: the secant is answering a question about an interval, not about a point.
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