The fundamental definition of derivative as instantaneous rate of change. This limit is what makes calculus possible.
Speedometer reading — Instantaneous velocity is derivative of position.
Stock price changes — Rate of price change at any moment.
Population growth rate — Instantaneous growth at any time.
Chemical reactions — Reaction rate as derivative of concentration.
Physics — Acceleration as derivative of velocity.
Economics — Marginal cost as derivative of total cost.
Medicine — Rate of drug absorption in bloodstream.
Engineering — Stress rate in material deformation.
Climate — Rate of temperature change.
Finance — Sensitivity of option price to underlying (Delta).
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