Differentiation Essentials for Edexcel A-Level Pure Mathematics | 爱德思A-Level纯数学：微分核心与考试技巧

📚 Differentiation Essentials for Edexcel A-Level Pure Mathematics | 爱德思A-Level纯数学：微分核心与考试技巧

Differentiation is one of the most heavily assessed topics in Edexcel A-Level Pure Mathematics. It underpins applications such as optimisation, curve sketching, rates of change and kinematics. This revision guide brings together the key definitions, rules and exam techniques you need to answer both routine and problem-solving questions with confidence.

微分是爱德思A-Level纯数学中考查最频繁的主题之一，它是优化问题、曲线作图、变化率与运动学等应用的基础。本复习指南汇总了关键定义、求导法则与考试技巧，帮助你自信应对常规题与综合题。

1. Introduction to Differentiation | 微分简介

In Edexcel Pure Maths, differentiation measures the instantaneous rate of change of a function y = f(x). The derivative is written as f ‘(x) or dy/dx. Geometrically, it gives the gradient of the tangent to the curve at any point.

在爱德思纯数学中，微分用于度量函数 y = f(x) 的瞬时变化率。导数写作 f ‘(x) 或 dy/dx。从几何上看，它给出曲线上任意一点切线的斜率。

If y = f(x), the derivative defines a new function that gives the gradient at each x-value. You are expected to interpret the derivative in both graphical and real-world contexts, including velocity as the rate of change of displacement.

若 y = f(x)，导数定义了一个新函数，给出每个 x 值处的斜率。你需要能够在图形和实际情境中解释导数，例如速度就是位移的变化率。

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

The derivative from first principles is defined by the limit of the difference quotient:

从第一性原理出发，导数定义为差商的极限：

f ‘(x) = limₕ→0 [f(x+h) − f(x)] / h

You may be asked to prove the derivative of a simple function such as f(x) = x² or f(x) = 1/x using this definition. Always show the expansion, cancellation and limiting process clearly.

考试可能会要求你用该定义证明简单函数的导数，如 f(x) = x² 或 f(x) = 1/x。解答时应清楚展示展开、约分与取极限的过程。

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h², so the difference quotient becomes (2xh + h²)/h = 2x + h, which tends to 2x as h tends to 0.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²，因此差商变为 (2xh + h²)/h = 2x + h，当 h 趋于 0 时极限为 2x。

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3. Standard Derivatives and Rules | 基本导数与求导法则

Learn the standard derivatives shown in the table below:

熟记下表列出的基本导数：

f(x)	f ‘(x)
xⁿ	n xⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec² x

The power rule d/dx (xⁿ) = n xⁿ⁻¹ is valid for all real n, including negative and fractional powers. Rewrite surds and reciprocals as powers before differentiating.

幂函数求导法则 d/dx (xⁿ) = n xⁿ⁻¹ 对所有实数 n 均成立，包括负数与分数次幂。求导前应先将根式与倒数改写为幂的形式。

For example, rewrite 1/x² as x⁻² and √x as x¹/², then apply the power rule: d/dx (1/x²) = −2x⁻³ and d/dx (√x) = ½ x⁻¹/².

例如，将 1/x² 改写为 x⁻²，将 √x 改写为 x¹/²，然后应用幂函数求导法则：d/dx (1/x²) = −2x⁻³，d/dx (√x) = ½ x⁻¹/²。

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4. Product, Quotient and Chain Rules | 乘法、除法与链式法则

Three rules handle combinations of functions. The product rule is:

三个法则用于处理函数的组合。乘法法则为：

d/dx [u v] = u’ v + u v’

The chain rule is:

链式法则为：

dy/dx = (dy/du) × (du/dx)

The quotient rule is:

商法则为：

d/dx [u/v] = (u’ v − u v’) / v²

In the exam, clearly identify u and v, find their derivatives separately, then substitute. For the chain rule, choose an inner function u carefully, especially for trigonometric, exponential and logarithmic forms.

考试中应明确写出 u 和 v，分别求导后再代入。使用链式法则时，要仔细选择内层函数 u，尤其是三角、指数与对数形式。

For instance, if y = e²ˣ, let u = 2x, so du/dx = 2 and dy/du = eᵘ. Therefore dy/dx = 2e²ˣ. Similarly, d/dx [ln(3x+1)] = 3/(3x+1).

例如，若 y = e²ˣ，令 u = 2x，则 du/dx = 2，dy/du = eᵘ。因此 dy/dx = 2e²ˣ。类似地，d/dx [ln(3x+1)] = 3/(3x+1)。

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5. Second Derivatives and Nature of Stationary Points | 二阶导数与驻点性质

The second derivative f ”(x) or d²y/dx² measures the rate of change of the gradient. It is used to classify stationary points.

二阶导数 f ”(x) 或 d²y/dx² 度量斜率的变化率，用于判断驻点性质。

If f ‘(a) = 0 and f ”(a) > 0, the point is a local minimum. If f ‘(a) = 0 and f ”(a) < 0, it is a local maximum. If f ”(a) = 0, use a sign change test on f ‘(x) or consider higher derivatives.

若 f ‘(a) = 0 且 f ”(a) > 0，则该点为局部极小值点；若 f ‘(a) = 0 且 f ”(a) < 0，则为局部极大值点。若 f ”(a) = 0，可对 f ‘(x) 做符号变化检验或考虑更高阶导数。

Stationary points occur where f ‘(x) = 0. After finding x-values, substitute into the original function to obtain the corresponding y-coordinates. Always present your classification clearly.

驻点出现在 f ‘(x) = 0 处。求出 x 值后，代入原函数得到相应的 y 坐标。分类时务必表述清晰。

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6. Tangents and Normals | 切线与法线

The gradient of the tangent at x = a is m = f ‘(a). Hence the tangent equation is:

在 x = a 处切线的斜率为 m = f ‘(a)，因此切线方程为：

y − f(a) = f ‘(a) (x − a)

The normal is perpendicular to the tangent, so its gradient is −1 / f ‘(a), giving:

法线与切线垂直，其斜率为 −1 / f ‘(a)，方程为：

y − f(a) = (−1 / f ‘(a)) (x − a)

Be careful when the tangent

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