Math & Statistics · 导数（Derivative） · Updated 2026-10-25
What Is Derivative?
Definition
The derivative f'(x)=lim(Δx→0)[f(x+Δx)−f(x)]/Δx is the tangent slope geometrically, instantaneous velocity physically, and the marginal (cost/revenue) economically. Differentiability implies continuity but not conversely (|x| has a kink at 0). Rules: sums split; products and quotients have formulas; composites use the chain rule (f(g(x)))'=f'(g)·g'(x). Optimization (max profit, min cost) classically sets the derivative to zero to find critical points.
Formula
f'(x) = lim(Δx→0) [f(x+Δx)−f(x)] / Δx
Worked Example
Profit π(q)=100q−q²−200; setting π'(q)=100−2q=0 gives q=50 as the maximum — "marginal revenue equals marginal cost" is exactly the zero-derivative condition.
Common Misconception
Reading f'(x0) as f(x0+1)−f(x0) — that is an average rate (secant slope); the derivative is the Δx→0 limit. And continuous functions are not all differentiable: |x| and step functions are continuous yet fail at kinks and jumps.
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