Mastering A-Level Edexcel Differentiation: Rules, Tangents and Optimisation | 掌握 A-Level Edexcel 微分：法则、切线与优化

📚 Mastering A-Level Edexcel Differentiation: Rules, Tangents and Optimisation | 掌握 A-Level Edexcel 微分：法则、切线与优化

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It underpins problems on gradients, tangents, stationary points, optimisation and rates of change. This article consolidates the core rules, standard derivatives and common exam applications you need to master.

微分是 Edexcel A-Level 数学中考查最频繁的主题之一。它是求解梯度、切线、驻点、优化和变化率等问题的基础。本文将整合你需要掌握的核心法则、标准导数以及常见考试应用。

1. The Big Idea of Differentiation | 微分的基本思想

Differentiation measures the instantaneous rate of change of a function. In graphical terms, the derivative f'(x) gives the gradient of the tangent to the curve y = f(x) at any point x. You should interpret f'(a) as the slope of the curve at x = a, not the slope of a chord between two distant points.

微分衡量函数的瞬时变化率。从图形上看，导数 f'(x) 给出曲线 y = f(x) 在任意点 x 处切线的梯度。你应当将 f'(a) 理解为曲线在 x = a 处的斜率，而不是两个远点之间弦的斜率。

f'(x) = limₕ→₀ [f(x+h) − f(x)] / h

This limit definition is the formal foundation and is tested in ‘differentiation from first principles’ questions. The notation dy/dx, f'(x) and d/dx [f(x)] all represent the same derivative.

该极限定义是形式化基础，也是“从第一原理求导”问题的考点。记号 dy/dx、f'(x) 和 d/dx [f(x)] 都表示同一个导数。

2. First Principles: The Limit Definition | 第一原理：极限定义

To differentiate from first principles, start with the limit of the difference quotient. For f(x) = x², you expand (x+h)², simplify the numerator, cancel h and then let h tend to 0. You must show each algebraic step clearly in the exam.

从第一原理求导时，从差商的极限开始。对于 f(x) = x²，你需要展开 (x+h)²，化简分子，约去 h，然后令 h 趋于 0。考试中必须清晰地展示每一步代数过程。

f'(x) = limₕ→₀ ((x+h)² − x²) / h = limₕ→₀ (2xh + h²) / h = 2x

The result f'(x) = 2x is a classic mark scheme requirement for Edexcel Paper 1. It is not enough to write the answer; examiners expect to see the limit process, expansion and cancellation of h.

结果 f'(x) = 2x 是 Edexcel 卷一评分标准中常见的得分点。仅写出答案是不够的，阅卷人期望看到极限过程、展开以及约去 h 的步骤。

For a general power xⁿ, the first principles expansion uses the binomial theorem, but the limiting idea remains the same. Practise writing the difference quotient for x² and x³ before attempting more abstract proofs.

对于一般幂 xⁿ，第一原理展开会使用二项式定理，但极限思想保持不变。在尝试更抽象的证明之前，先练习写出 x² 和 x³ 的差商。

3. Standard Derivatives and the Power Rule | 标准导数与幂法则

The power rule states that if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹ for any real n. It works for positive integers, negative powers and fractional powers. Rewrite roots and reciprocals as powers before differentiating.

幂法则指出，如果 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 为任意实数。它适用于正整数、负指数和分数指数。求导前要先将根式和倒数改写为幂的形式。

d/dx (xⁿ) = n xⁿ⁻¹

• f(x) = x⁵ → f'(x) = 5x⁴
• f(x) = 1/x² = x⁻² → f'(x) = −2x⁻³
• f(x) = √x → f'(x) = 1 / (2√x)

For roots such as ∛x or 1/√x, first convert to x^⅓ or x^−½. The same power rule then applies, but many students lose marks by forgetting to subtract 1 from a fractional exponent.

对于 ∛x 或 1/√x 这样的根式，先转化为 x^⅓ 或 x^−

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