📚 Differentiation & Integration Techniques | 爱德思 A-Level 纯数学：微分与积分技巧
This revision guide covers the core differentiation and integration techniques required for Edexcel A-Level Pure Mathematics. You will see how to apply the chain rule, product rule, quotient rule, parametric and implicit differentiation, as well as standard integration, substitution, integration by parts, and area problems.
本复习指南涵盖爱德思 A-Level 纯数学要求的微积分核心技巧。你将学习如何运用链式法则、乘积法则、商法则、参数方程微分和隐函数微分，以及基本积分、换元积分、分部积分和面积问题。
1. The Chain Rule | 链式法则
The chain rule is used to differentiate a composite function y = f(g(x)). If u = g(x) and y = f(u), then dy/dx = dy/du × du/dx.
链式法则用于对复合函数 y = f(g(x)) 求导。若 u = g(x) 且 y = f(u)，则 dy/dx = dy/du × du/dx。
For example, differentiate y = (3x² + 5)⁴. Let u = 3x² + 5, then y = u⁴, dy/du = 4u³, du/dx = 6x, so dy/dx = 24x(3x² + 5)³.
例如，对 y = (3x² + 5)⁴ 求导。设 u = 3x² + 5，则 y = u⁴，dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。
In Edexcel papers, the chain rule often appears combined with trigonometric, exponential, and logarithmic functions. Always identify the inner function first.
在爱德思考试中，链式法则常与三角函数、指数函数和对数函数结合考查。务必先识别内层函数。
2. Product and Quotient Rules | 乘积法则与商法则
For a product y = u(x)v(x), the product rule states dy/dx = u’v + uv’. For a quotient y = u/v, the quotient rule states dy/dx = (u’v – uv’) / v².
对于乘积 y = u(x)v(x)，乘积法则为 dy/dx = u’v + uv’。对于商 y = u/v，商法则为 dy/dx = (u’v – uv’) / v²。
Example: y = x² sin x. Let u = x², v = sin x. Then u’ = 2x, v’ = cos x, so dy/dx = 2x sin x + x² cos x.
示例：y = x² sin x。设 u = x²，v = sin x。则 u’ = 2x，v’ = cos x，所以 dy/dx = 2x sin x + x² cos x。
- Product rule is usually applied when two different functions are multiplied.
- 乘积法则通常用于两个不同类型的函数相乘。
- Quotient rule is needed for rational expressions such as y = eˣ / (x + 1).
- 商法则适用于有理表达式，如 y = eˣ / (x + 1)。
3. Parametric Differentiation | 参数方程微分
When x = f(t) and y = g(t), the derivative dy/dx is given by dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
当 x = f(t) 且 y = g(t) 时，导数 dy/dx 由 dy/dx = (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。
For example, if x = t² and y = t³, then dx/dt = 2t, dy/dt = 3t², so dy/dx = (3t²)/(2t) = 3t/2.
例如，若 x = t² 且 y = t³，则 dx/dt = 2t，dy/dt = 3t²，因此 dy/dx = (3t²)/(2t) = 3t/2。
To find the second derivative d²y/dx², use d²y/dx² = d/dx(dy/dx) = d/dt(dy/dx) ÷ dx/dt.
求二阶导数 d²y/dx² 时，使用 d²y/dx² = d/dx(dy/dx) = d/dt(dy/dx) ÷ dx/dt。
4. Implicit Differentiation | 隐函数微分
When an equation mixes x and y without an explicit y = f(x) form, differentiate both sides with respect to x, treating y as a function of x. Each time you differentiate a term containing y, multiply by dy/dx.
当方程混合 x 和 y 且未给出显式 y = f(x) 形式时，对等式两边关于 x 求导，并将 y 视为 x 的函数。每次对含 y 的项求导后，要乘以 dy/dx。
Example: x² + y² = 25. Differentiating gives 2x + 2y(dy/dx) = 0, so dy/dx = -x/y.
示例：x² + y² = 25。求导得 2x + 2y(dy/dx) = 0，因此 dy/dx = -x/y。
This technique is essential for finding gradients of tangents to curves defined implicitly, such as circles and ellipses.
这一技巧对于求隐式定义的曲线（如圆和椭圆）的切线斜率至关重要。
5. Standard Integration and Reverse Differentiation | 基本积分与逆微分
Integration reverses differentiation. The basic result is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1, and ∫ 1/x dx = ln|x| + C.
积分是微分的逆运算。基本结果为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ -1），以及 ∫ 1/x dx = ln|x| + C。
For trigonometric functions, ∫ sin x dx = -cos x + C and ∫ cos x dx = sin x + C. For exponentials, ∫ eˣ dx = eˣ + C.
对于三角函数，∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。对于指数函数，∫ eˣ dx = eˣ + C。
Always add the constant of integration C for indefinite integrals. In definite integrals, apply limits and subtract.
不定积分务必加上积分常数 C。定积分则代入上下限并相减。
6. Integration by Substitution | 换元积分法
The substitution method simplifies an integral by replacing a complicated expression with a new variable u. If u = g(x), then du/dx = g'(x), so dx = du / g'(x).
换元积分法通过用新变量 u 替换复杂表达式来简化积分。若 u = g(x)，则 du/dx = g'(x)，因此 dx = du / g'(x)。
Example: evaluate ∫ 2x(x² + 1)⁴ dx. Let u = x² + 1, then du/dx = 2x, so dx = du/(2x). The integral becomes ∫ u⁴ du = u⁵/5 + C = (x² + 1)⁵/5 + C.
示例：计算 ∫ 2x(x² + 1)⁴ dx。设 u = x² + 1，则 du/dx = 2x，因此 dx = du/(2x)。积分变为 ∫ u⁴ du = u⁵/5 + C = (x² + 1)⁵/5 + C。
In Edexcel exams, look for a function and its derivative appearing together, such as ∫ sin x cos²x dx or ∫ x eˣ² dx.
在爱德思考试中，注意被积函数中是否同时出现一个函数及其导数，例如 ∫ sin x cos²x dx 或 ∫ x eˣ² dx。
7. Integration by Parts | 分部积分法
Integration by parts is based on the product rule for differentiation. The formula is ∫ u dv = uv – ∫ v du.
分部积分法基于微分的乘积法则。公式为 ∫ u dv = uv – ∫ v du。
Choose u according to the LIATE order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Differentiate u and integrate dv.
选择 u 时按 LIATE 顺序：对数函数、反三角函数、代数函数、三角函数、指数函数。对 u 求导，对 dv 积分。
Example: ∫ x eˣ dx. Let u = x, dv = eˣ dx. Then du = dx, v = eˣ. So ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C.
示例：∫ x eˣ dx。设 u = x，dv = eˣ dx。则 du = dx，v = eˣ。因此 ∫ x eˣ dx = x eˣ – ∫ eˣ dx = x eˣ – eˣ + C。
For definite integrals, evaluate uv at the limits and subtract the integral of v du between the same limits.
对于定积分，先在上下限处计算 uv，再减去 v du 在同一上下限的积分。
8. Area Under a Curve and Between Curves | 曲线下面积与曲线间面积
The area under y = f(x) from x = a to x = b is given by ∫ₐᵇ f(x) dx, provided f(x) ≥ 0 on [a, b].
曲线 y = f(x) 从 x = a 到 x = b 下方的面积由 ∫ₐᵇ f(x) dx 给出，前提是 f(x) ≥ 0 在 [a, b] 上。
If the curve lies below the x-axis, the integral is negative, so take the absolute value or split the interval.
若曲线位于 x 轴下方，积分为负，因此取绝对值或拆分区间。
For the area between two curves y = f(x) and y = g(x), use ∫ₐᵇ [f(x) – g(x)] dx, where f(x) ≥ g(x) on [a, b].
对于两条曲线 y = f(x) 与 y = g(x) 之间的面积，使用 ∫ₐᵇ [f(x) – g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。
Always find points of intersection to determine the limits of integration.
务必求出交点以确定积分上下限。
9. Trapezium Rule and Numerical Integration | 梯形法则与数值积分
The trapezium rule approxim
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