Mastering Differentiation: Core Techniques for Edexcel A-Level Pure Mathematics | 掌握微分：Edexcel A-Level 纯数学核心技巧

📚 Mastering Differentiation: Core Techniques for Edexcel A-Level Pure Mathematics | 掌握微分：Edexcel A-Level 纯数学核心技巧

Differentiation is one of the most heavily examined threads in the Edexcel A-Level Mathematics specification. A reliable command of derivative rules, notation, and geometric interpretation gives you immediate access to marks on rates of change, tangents, normals, optimisation, and curve sketching. This guide walks through the core Pure Mathematics techniques in a structured, exam-focused way.

微积分中的微分是 Edexcel A-Level 数学大纲中考查最密集的核心内容之一。扎实掌握求导法则、符号体系和几何含义，能让你在变化率、切线与法线、最优化以及函数图像分析等题型中迅速得分。本文按考点结构梳理纯数学中的核心微分技巧。

1. What Differentiation Means | 微分的意义与记号

In pure mathematics, differentiation finds the instantaneous rate of change of a function. For a curve y = f(x), the derivative f'(x) is the gradient of the tangent at a point.

在纯数学中，微分用于求函数的瞬时变化率。对于曲线 y = f(x)，导数 f'(x) 表示某一点处切线的斜率。

The two most common Edexcel notations are dy/dx and f'(x). The operator d/dx means differentiate with respect to x, so dy/dx = f'(x).

Edexcel 考试中最常见的两种记号是 dy/dx 和 f'(x)。算子 d/dx 表示对 x 求导，因此 dy/dx = f'(x)。

dy/dx = lim (Δx → 0) Δy/Δx

This ratio measures the limiting gradient of the secant line as Δx approaches zero. On a graph, the derivative is the slope of the tangent line.

该比值表示当 Δx 趋近于零时割线斜率的极限。在图像上，导数就是切线的斜率。

2. Differentiation from First Principles | 从第一性原理求导

The formal definition of the derivative is the limit of the difference quotient:

导数的正式定义是差商的极限：

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

This is often tested as a standalone Edexcel question, especially for simple polynomials such as x² or x³.

这经常作为 Edexcel 独立考题出现，尤其针对 x² 或 x³ 等简单多项式。

For f(x) = x², expand f(x + h) = (x + h)² = x² + 2xh + h². Subtracting f(x) gives 2xh + h², dividing by h gives 2x + h. Let h → 0 to obtain 2x.

对于 f(x) = x²，展开 f(x + h) = (x + h)² = x² + 2xh + h²。减去 f(x) 得到 2xh + h²，除以 h 得到 2x + h。令 h → 0 得 2x。

f'(x) = 2x

Remember that the h term must cancel before taking the limit; otherwise the expression is undefined at h = 0.

记住在取极限前 h 必须能约去，否则表达式在 h = 0 处无定义。

3. The Power Rule and Index Manipulation | 幂函数法则与指数处理

For any real constant n, the power rule states:

对于任意实常数 n，幂函数求导法则为：

d/dx (xⁿ) = n xⁿ⁻¹

This rule works for negative and fractional powers, which makes it essential for roots and reciprocals.

该法则同样适用于负指数和分数指数，因此对于根式与倒数形式十分关键。

• d/dx (x⁵) = 5x⁴
• d/dx (x⁻¹) = −x⁻² = −1/x²
• d/dx (√x) = d/dx (x¹/²) = ½ x⁻¹/² = 1/(2√x)

中文对应： d/dx (x⁵) = 5x⁴；d/dx (x⁻¹) = −x⁻² = −1/x²；d/dx (√x) = d/dx (x¹/²)

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