The Edexcel A-Level Calculus Toolkit: Differentiation and Integration Skills | Edexcel A-Level 微积分工具箱：微分与积分核心技巧

📚 The Edexcel A-Level Calculus Toolkit: Differentiation and Integration Skills | Edexcel A-Level 微积分工具箱：微分与积分核心技巧

Calculus is at the heart of the Edexcel A-Level Mathematics specification, appearing in both Pure Paper 1 and Pure Paper 2. This revision guide brings together the differentiation and integration techniques most frequently examined, from first principles to differential equations, with worked-style explanations of each method.

微积分是 Edexcel A-Level 数学考试的核心内容，贯穿 Pure Paper 1 与 Pure Paper 2。本复习指南将最常考查的微分与积分技巧整合在一起，从第一原理到微分方程，逐一说明每种方法，并给出解题思路。

1. Differentiation from First Principles | 从第一原理出发的微分

The first principles definition is used in Paper 1 to prove the derivative of simple functions such as x² and x³. You must expand f(x+h), simplify the difference quotient, and then let h approach 0.

第一原理定义常用于 Paper 1 中证明 x²、x³ 等简单函数的导数。你需要展开 f(x+h)，化简差商，然后令 h 趋近于 0。

f'(x) = limₕ→₀ [f(x+h) − f(x)] / h

For f(x)=x², expand (x+h)² = x² + 2xh + h², subtract x², divide by h, and the limit gives 2x.

以 f(x)=x² 为例，展开 (x+h)² = x² + 2xh + h²，减去 x²，再除以 h，取极限后得到 2x。

2. Standard Derivatives and the Chain Rule | 标准导数与链式法则

Know the standard results: d/dx of xⁿ is nxⁿ⁻¹, d/dx of eˣ is eˣ, d/dx of ln x is 1/x, d/dx of sin x is cos x, d/dx of cos x is −sin x, and d/dx of tan x is sec²x.

牢记标准导数：xⁿ 的导数为 nxⁿ⁻¹，eˣ 的导数为 eˣ，ln x 的导数为 1/x，sin x 的导数为 cos x，cos x 的导数为 −sin x，tan x 的导数为 sec²x。

The chain rule is used when a function is composite. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

当函数为复合函数时使用链式法则。若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。

d/dx [f(g(x))] = f'(g(x)) × g'(x)

For example, y = sin(3x) gives dy/dx = 3 cos(3x), and y = e²ˣ gives dy/dx = 2 e²ˣ.

例如，y = sin(3x) 的导数为 dy/dx = 3 cos(3x)，y = e²ˣ 的导数为 dy/dx = 2 e²ˣ。

3. Product, Quotient and Second Derivatives | 乘积法则、商法则与二阶导数

The product rule states that if y = u v, then dy/dx = u dv/dx + v du/dx. The quotient rule states that if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².

乘积法则指出，若 y = u v，则 dy/dx = u dv/dx + v du/dx。商法则指出，若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。

Product rule: d/dx (uv) = u dv/dx + v du/dx

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

Avoid cancelling terms incorrectly in the quotient rule; the denominator is always v². The second derivative f”(x) is found by differentiating f'(x) again and is used to classify stationary points.

使用商法则时切勿错误地约分；分母始终为 v²。二阶导数 f”(x) 是对 f'(x) 再次求导得到的，它用于判断驻点类型。

4. Tangents, Normals

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）