Question A smartwatch logs distance along a lakeside loop. The odometer is a running total — how far you've gone. The little pace number flickering every second? That's speaking a different language. Derivatives live in pace language: how fast a quantity is changing at a moment. Formally, the derivative at a is the limit of average rates (f(a+h) − f(a))/h as h approaches 0 — when that limit exists.
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State the limit definition of f'(a) Estimate f'(a) from a shrinking difference-quotient table Connect derivative, tangent slope, and instantaneous rate Give a continuous-but-not-differentiable example
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Odometer → totals Pace readout → rates Which is derivative? f'(a) = lim h→0 (f(a+h)−f(a))/h
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