Edexcel A-Level Pure Maths: Differentiation from First Principles to Parametric and Implicit Methods | 爱德思A-Level纯数学：从第一原理到参数与隐函数的微分方法

📚 Edexcel A-Level Pure Maths: Differentiation from First Principles to Parametric and Implicit Methods | 爱德思A-Level纯数学：从第一原理到参数与隐函数的微分方法

Differentiation is one of the core pillars of the Edexcel A-Level Pure Mathematics specification. It appears not only as a standalone skill but also inside integration, kinematics, optimisation and proof questions. This revision guide builds the topic from first principles, then moves through standard derivatives, chain, product, quotient, implicit and parametric methods, and finishes with applied contexts such as tangents, normals and rates of change.

微分是爱德思A-Level纯数学考试的核心支柱之一。它不仅作为独立技能考查，还会出现在积分、运动学、优化和证明题中。本复习指南从第一原理出发，逐步讲解标准导数、链式法则、乘法法则、商法则、隐函数和参数微分方法，最后延伸到切线、法线和变化率等应用情境。

1. First Principles | 第一原理

Differentiation from first principles uses the limit of a gradient chord as the interval tends to zero. For a function f(x), the derivative is defined by f'(x) = lim (h → 0) [f(x+h) − f(x)] / h.

第一原理求导使用弦斜率的极限，使区间趋于零。对于函数 f(x)，导数定义为 f'(x) = lim（h → 0）[f(x+h) − f(x)] / h。

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

You will often be asked to prove the derivative of x² or x³ from first principles. Expand the bracket, simplify, and only then let h → 0.

考试中常要求从第一原理证明 x² 或 x³ 的导数。先展开括号、化简，然后才令 h → 0。

2. Standard Derivatives and Notation | 标准导数与记号

Edexcel papers expect fluent use of dy/dx, f'(x), d/dx notation and the standard results for powers, exponentials, logarithms and trigonometric functions.

爱德思试卷要求熟练使用 dy/dx、f'(x)、d/dx 记号，以及幂函数、指数、对数和三角函数的导数公式。

• d/dx (xⁿ) = n xⁿ⁻¹ | 中文：d/dx（xⁿ）= n xⁿ⁻¹
• d/dx (eˣ) = eˣ | 中文：d/dx（eˣ）= eˣ
• d/dx (ln x) = 1/x，x > 0 | 中文：d/dx（ln x）= 1/x，x > 0
• d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x | 中文：d/dx（sin x）= cos x，d/dx（cos x）= −sin x，d/dx（tan x）= sec² x

Remember that a constant multiple and a sum can be differentiated term by term: if y = a u(x) + b v(x), then dy/dx = a u'(x) + b v'(x).

记住常数倍与和式可以逐项求导：若 y = a u(x) + b v(x)，则 dy/dx = a u'(x) + b v'(x)。

3. Chain Rule | 链式法则

The chain rule is used for composite functions y = f(g(x)). It states dy/dx = dy/du × du/dx. In simple form, if y = [g(x)]ⁿ, then dy/dx = n[g(x)]ⁿ⁻¹ g'(x).

链式法则用于复合函数 y = f(g(x))。它表示为 dy/dx = dy/du × du/dx。简单形式下，若 y = [g(x)]ⁿ，则 dy/dx = n[g(x)]ⁿ⁻¹ g'(x)。

For y = (3x² + 5)⁴, set u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 24x(3x² + 5)³.

例如 y = (3x² + 5)⁴，令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。

Watch for chain rule combined with exponentials, logarithms and trig, such as d/dx [e^(2x³)] = 6x² e^(2x³).

注意链式法则与指数、对数和三角函数结合，例如 d/dx[e^(2x³)] = 6x² e^(2x³)。

4. Product Rule | 乘法法则

When y = u(x)v(x), the product rule states dy/dx = u’v + uv’. Select u and v clearly, differentiate each, then substitute.

当 y = u(x)v(x) 时，乘法法则为 dy/dx = u’v + uv’。应清晰选取 u 和 v，分别求导后再代入。

dy/dx =

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