PDF Joiner (4) Topic 201: Edexcel A-Level Maths Differentiation & Integration | PDF合并(4)专题201：Edexcel A-Level数学微分与积分

📚 PDF Joiner (4) Topic 201: Edexcel A-Level Maths Differentiation & Integration | PDF合并(4)专题201：Edexcel A-Level数学微分与积分

In this revision guide, we focus on the essential pure mathematics techniques of differentiation and integration. These skills appear throughout Edexcel A-Level Maths papers, from basic curve sketching to advanced differential equations. Mastery of the rules, applications and common exam pitfalls is crucial for achieving top grades.

本复习指南聚焦纯数学中的微分与积分核心技巧。这些技能贯穿 Edexcel A-Level 数学试卷，从基础曲线描绘到高级微分方程。掌握法则、应用和常见考试陷阱对于取得高分至关重要。

---

1. Key Differentiation Rules | 微分关键法则

Differentiation measures the rate of change of a function. For any power function y = xⁿ, the derivative is found by multiplying by the power and reducing the power by one.

微分衡量函数的变化率。对于任意幂函数 y = xⁿ，其导数通过乘以指数并将指数减一求得。

d/dx (xⁿ) = n xⁿ⁻¹

For example, if y = x⁵, then dy/dx = 5x⁴. This rule works for all real powers, including negative and fractional powers after rewriting roots and reciprocals.

例如，若 y = x⁵，则 dy/dx = 5x⁴。该法则适用于所有实数幂，包括将根式和倒数改写后的负指数和分数指数。

Constants differentiate to zero, and the derivative of a sum is the sum of the derivatives. Thus d/dx (3x² + 2x – 7) = 6x + 2.

常数项的导数为零，和的导数等于导数的和。因此 d/dx (3x² + 2x – 7) = 6x + 2。

---

2. Chain, Product & Quotient Rules | 链式法则、乘积法则与商法则

The chain rule differentiates composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). This is often written as dy/dx = dy/du × du/dx.

链式法则用于复合函数求导。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。常写作 dy/dx = dy/du × du/dx。

y = (3x² + 1)⁵ ⇒ dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴

The product rule states that for y = u v, the derivative is dy/dx = u dv/dx + v du/dx. The quotient rule for y = u/v is dy/dx = (v du/dx – u dv/dx) / v².

乘积法则指出，对于 y = u v，导数为 dy/dx = u dv/dx + v du/dx。商法则对于 y = u/v，导数为 dy/dx = (v du/dx – u dv/dx) / v²。

These rules are essential when differentiating expressions like x² sin x or (x+1)/(x-2). Always simplify before or after differentiating where possible.

这些法则在求导如 x² sin x 或 (x+1)/(x-2) 的表达式时至关重要。尽可能在求导前或求导后进行化简。

---

3. Second Derivatives & Stationary Points | 二阶导数与驻点

The second derivative, written d²y/dx² or f”(x), measures the rate of change of the gradient. It helps classify stationary points where dy/dx = 0.

二阶导数，写作 d²y/dx² 或 f”(x)，衡量梯度的变化率。它帮助对 dy/dx = 0 的驻点进行分类。

If f”(a) > 0, the stationary point at x = a is a local minimum. If f”(a) < 0, it is a local maximum. If f''(a) = 0, further investigation is needed.

若 f”(a) > 0，则 x = a 处的驻点为局部极小值。若 f”(a) < 0，则为局部极大值。若 f''(a) = 0，则需要进一步判断。

For example, y = x³ – 3x has dy/dx = 3x² – 3. Setting dy/dx = 0 gives x = ±1. The second derivative d²y/dx² = 6x shows x = 1 is a minimum and x = -1 is a maximum.

例如，y = x³ – 3x 的 dy/dx = 3x² – 3。令 dy/dx = 0 得 x = ±1。二阶导数 d²y/dx² = 6x 表明 x = 1 为极小值，x = -1 为极大值。

---

4. Integration as Reverse Differentiation | 积分作为微分的逆运算

Indefinite integration reverses differentiation. For any power n ≠ -1, the integral of xⁿ with respect to x is xⁿ⁺¹/(n+1) plus a constant of integration C.

不定积分是微分的逆运算。对于任意 n ≠ -1，xⁿ 对 x 的积分为 xⁿ⁺¹/(n+1) 加上积分常数 C。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

Never forget the constant C in indefinite integrals. It appears because differentiation of any constant gives zero, so infinitely many functions share the same derivative.

切勿忘记不定积分中的常数 C。它存在的原因是因为任意常数的导数为零，所以无穷多个函数拥有相同的导数。

Basic forms include ∫ cos x dx = sin x + C, ∫ sin x dx = -cos x + C and ∫ eˣ dx = eˣ + C. These must be memorised for the exam.

基本形式包括 ∫ cos x dx = sin x + C、∫ sin x dx = -cos x + C 和 ∫ eˣ dx = eˣ + C。这些必须牢记以应对考试。

---

5. Definite Integration & Area Under Curves | 定积分与曲线下方面积

A definite integral has upper and lower limits: ∫ₐᵇ f(x) dx. It gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b.

定积分具有上下限：∫ₐᵇ f(x) dx。它给出曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有符号面积。

Evaluate by finding the antiderivative F(x), then computing F(b) – F(a). Areas below the x-axis are negative, so split the interval where the curve crosses the axis.

计算方法是先求原函数 F(x)，再计算 F(b) – F(a)。x 轴下方的面积为负，因此在曲线穿过 x 轴处需拆分区间。

For example, ∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8. This represents the exact area under the line y = 2x between x = 1 and x = 3.

例如，∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8。这表示直线 y = 2x 在 x = 1 与 x = 3 之间的精确面积。

---

6. Integration by Substitution | 换元积分法

Integration by substitution is the reverse chain rule. It transforms a complicated integral into a simpler one by changing the variable, usually letting u equal an inner function.

换元积分法是链式法则的逆运算。它通过变量替换将复杂积分化为简单积分，通常令 u 等于内层函数。

For ∫ 2x(x²+1)⁴ dx, let u = x² + 1, so du/dx = 2x and du = 2x dx. The integral becomes ∫ u⁴ du = u⁵/5 + C = (x²+1)⁵/5 + C.

对于 ∫ 2x(x²+1)⁴ dx，令 u = x² + 1，则 du/dx = 2x 且 du = 2x dx。积分变为 ∫ u⁴ du = u⁵/5 + C = (x²+1)⁵/5 + C。

For definite integrals, change the limits to u-values or return to the original variable before applying the original limits. Both methods are acceptable but be consistent.

对于定积分，可将上下限转换为 u 值，或在代入原始上下限前换回原变量。两种方法均可，但必须保持一致。

---

7. Integration by Parts | 分部积分法

Integration by parts derives from the product rule. The formula is ∫ u dv = u v – ∫ v du, where u and dv are chosen carefully.

分部积分法源自乘积法则。公式为 ∫ u dv = u v – ∫ v du，其中 u 和 dv 需谨慎选择。

∫ u dv/dx dx = u v – ∫ v du/dx dx

A common choice is

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

更多咨询请联系16621398022（同微信）