Edexcel A-Level Pure Mathematics: Differentiation Rules and Applications | 爱德思A-Level纯数学：微分法则与应用

📚 Edexcel A-Level Pure Mathematics: Differentiation Rules and Applications | 爱德思A-Level纯数学：微分法则与应用

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It connects algebraic techniques with real-world rates of change, optimisation and curve analysis.

微分是爱德思A-Level数学中考查最频繁的主题之一。它把代数技巧与现实中的变化率、优化问题和曲线分析联系在一起。

This revision guide covers first principles, the main differentiation rules, implicit and parametric methods, connected rates and stationary points. Each section is written as an exam-focused summary of the techniques Edexcel candidates need to apply confidently.

本复习指南涵盖第一性原理、主要求导法则、隐函数与参数方法、相关变化率和驻点。每一节都以考试重点为导向，总结爱德思考生需要熟练运用的技巧。

1. Gradient and Derivative from First Principles | 梯度与第一性原理求导

The derivative f'(x) measures the instantaneous rate of change of a function at a point. From first principles, it is the limit of average gradients of chords as the interval tends to zero.

导数 f'(x) 度量函数在某一点的瞬时变化率。从第一性原理出发，它是当区间趋于零时弦的平均梯度的极限。

f'(x) = lim (h → 0) [f(x + h) – f(x)] / h

For example, differentiating f(x) = x² from first principles gives f'(x) = 2x, because the difference quotient simplifies to 2x + h and then h tends to zero.

例如，对 f(x) = x² 从第一性原理求导可得 f'(x) = 2x，因为差商可化简为 2x + h，然后令 h 趋于零。

In Edexcel questions, first-principles proofs are usually required only for simple polynomial terms, but the limit interpretation underpins every differentiation rule you will use.

在爱德思考试题中，通常只要求证明简单多项式项的第一性原理，但极限意义是你将使用的每一条微分法则的基础。

2. Standard Derivatives and the Power Rule | 常见导数与幂法则

The power rule states that if f(x) = xⁿ for any real constant n, then f'(x) = n xⁿ⁻¹. The rule is extended linearly: constants multiply derivatives, and sums differentiate term by term.

幂法则指出，如果 f(x) = xⁿ，其中 n 为任意实常数，则 f'(x) = n xⁿ⁻¹。该法则可线性推广：常数倍可直接提出，和可逐项求导。

d/dx (xⁿ) = n xⁿ⁻¹

You must be careful with negative and fractional powers. For instance, d/dx (x⁻²) =

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