Derivative as Slope
Skill goal: see the derivative as the limiting slope of secants.
Observe
The fixed gold point P sits at x = 1. The teal point Q slides along the curve. Watch what happens to the secant line’s slope as Q moves toward P.
Manipulate
Tip: drag the teal point Q along the curve, use the h slider, or press “Move Q closer” to animate h → 0.
Predict: Before you change it, what do you think will happen? As Q gets very close to P, what value does the secant slope approach?
See
h (distance PQ)
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Q = (1+h, f(1+h))
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Secant slope
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Exact f′(1)
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Explain
Why it works: The secant slope [f(1+h) − f(1)] / h is the average rate of change between P and Q. As h → 0, Q slides into P and the secant line settles onto the tangent line — its slope becomes the instantaneous rate of change, the derivative f′(1). That is why the gold tangent line fades in only when h is tiny.
Challenge
1. For f(x) = x² at x = 1, what is the exact derivative f′(1)?
2. Switch to f(x) = sin x. The derivative f′(1) = cos(1) ≈ ? (give 3 decimals)