Mastering Differentiation and Integration for Edexcel A-Level Pure Maths | 掌握Edexcel A-Level纯数学微积分核心方法

📚 Mastering Differentiation and Integration for Edexcel A-Level Pure Maths | 掌握Edexcel A-Level纯数学微积分核心方法

Differentiation and integration are the twin pillars of A-Level Pure Mathematics. In the Edexcel specification, these topics account for a large proportion of both Paper 1 and Paper 2, and they also appear in mechanics and statistics contexts. This article summarises the key rules, common techniques and exam-style applications you need to master.

微分与积分是 A-Level 纯数学的两大支柱。在 Edexcel 考试大纲中，这两个主题在 Paper 1 和 Paper 2 中占很大比例，并且在力学与统计背景下也会出现。本文总结你必须掌握的关键法则、常用技巧和考试型应用。

1. Gradient Functions and First Principles | 梯度函数与第一原理

The derivative of a function f(x) gives the gradient of the tangent to the curve y = f(x) at any point. The formal definition from first principles is based on a limit as h approaches zero.

函数 f(x) 的导数给出曲线 y = f(x) 在任意点处切线的斜率。根据第一原理的正式定义基于 h 趋近于零时的极限。

f'(x) = lim (h→0) [f(x+h) − f(x)] / h

For a polynomial term xⁿ, expanding (x+h)ⁿ shows that the derivative is n xⁿ⁻¹. You should be able to apply this result to any polynomial or rational power.

对于多项式项 xⁿ，展开 (x+h)ⁿ 可证明其导数为 n xⁿ⁻¹。你应能将这一结果应用于任意多项式或有理次幂。

If f(x) = x³, then f'(x) = 3x².

2. Differentiation Rules: Power, Chain, Product, Quotient | 微分法则：幂、链式、乘积、商

The Edexcel papers expect fluency in the power rule, chain rule, product rule and quotient rule. In composite functions, differentiate the outer function and multiply by the derivative of the inner function.

Edexcel 试卷要求熟练掌握幂法则、链式法则、乘积法则和商法则。对于复合函数，先对外层函数求导，再乘以内层函数的导数。

d/dx [f(g(x))] = f'(g(x)) g'(x)

For products and quotients, the order of terms matters. It is useful to set u and v clearly before applying the rules.

对于乘积和商，项的顺序很重要。在应用法则之前，建议先明确写出 u 和 v。

Product rule: d/dx (u v) = u’ v + u v’

Quotient rule: d/dx (u/v) = (u’ v − u v’) / v²

3. Derivatives of Exponentials, Logs and Trigonometric Functions | 指数、对数与三角函数的导数

The standard derivatives of eˣ, ln x, sin x, cos x and tan x must be memorised. When these functions have a linear inner function, multiply by its coefficient.

必须牢记 eˣ、ln x、sin x、cos x 和 tan x 的标准导数。当这些函数的内层是线性函数时，要乘以其系数。

For example, d/dx (e³ˣ) = 3e³ˣ and d/dx (sin 2x) = 2 cos 2x. Edexcel often tests these in context with the chain rule.

例如，d/dx (e³ˣ) = 3e³ˣ，d/dx (sin 2x) = 2 cos 2x。Edexcel 常结合链式法则考查这些内容。

4. Tangents, Normals and Stationary Points | 切线、法线与驻点

The tangent to a curve at x = a has gradient f'(a) and passes through the point (a, f(a)). The normal is perpendicular, so its gradient is −1/f'(a).

曲线在 x = a 处的切线斜率为 f'(a)，且经过点 (a, f(a))。法线垂直于切线，因此其斜率为 −1/f'(a)。

Tangent: y − f(a) = f'(a)(x − a)

Normal: y − f(a) = −(1/f'(a))(x − a)

Stationary points occur where dy/dx = 0. Their nature is determined using the second derivative or the sign change of dy/dx.

驻点出现在 dy/dx = 0 处。其性质可通过二阶导数或 dy/dx 的符号变化来判断。

5. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative d²y/dx² measures the rate of change of the gradient. If d²y/dx² > 0 at a stationary point, the point is

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