📚 Edexcel A-Level Maths: Core Differentiation and Integration Techniques | Edexcel A-Level 数学：核心微分与积分技巧
s. Mastering these methods is essential for tackling rates of change, optimisation, areas under curves, volumes of revolution and first-order differential equations. It is written for students working through mixed Edexcel A-Level practice files, including the set labelled pdfjoiner_(4)-156.
本复习指南涵盖 Edexcel A-Level 纯数试卷中最常见的核心微分与积分技巧。掌握这些方法对于解决变化率、最优化、曲线下面积、旋转体体积以及一阶微分方程问题至关重要。本指南适用于练习 Edexcel A-Level 混合习题的学生，包括编号为 pdfjoiner_(4)-156 的练习集。
1. Chain Rule | 链式法则
The chain rule is the most frequently used technique for differentiating composite functions, where one function is applied inside another. It appears in almost every paper, and it also underpins implicit differentiation, related rates of change and the reverse chain rule used in integration.
链式法则是对复合函数求导时最常用的技巧，其中一个函数嵌套在另一个函数内部。它几乎出现在每一份试卷中，同时也是隐函数求导、相关变化率以及积分中反向链式法则的基础。
In function notation, if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). In Leibniz notation the same idea becomes dy/dx = dy/du × du/dx, which shows the 'cancelling' erentiate the outer function first, then multiply by the derivative of the inner function.
用函数记号表示，若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。换成莱布尼茨记号，同一思想写作 dy/dx = dy/du × du/dx，这种写法直观显示出 du 的'约去'，并提醒你先对外层函数求导，再乘以内层函数的导数。
dy/dx = dy/du × du/dx
Typical Edexcel contexts include the following:
Edexcel 试卷中的典型情形包括以下几类：
- Powers of linear or non-linear brackets, such as (3x + 5)⁷ or (x² + 1)⁵.
- Exponentials of functions, such as e4x − 1 or ex².
- Logarithms of functions, such as ln(5x + 2) or ln(sin x).
- Trigonometric functions of non-linear arguments, such as sin(3x) or cos(x²).
- Square roots and reciprocal powers, such as √2x + 7 or 1/(x + 4)³.
ction is a fourth power, so dy/dx = 4(2x³ − 5)³ × 6x², which simplifies to 24x²(2x³ − 5)³.
例题：求 y = (2x³ − 5)⁴ 的导数。外层是四次幂，因此 dy/dx = 4(2x³ − 5)³ × 6x²，化简得 24x²(2x³ − 5)³。
The most common error is forgetting the inner derivative, giving 4(2x³ − 5)³ only. Whenever you differentiate anything with brackets, ask yourself: what is the derivative of the inside?
最常见的错误是漏掉内层导数，只写成 4(2x³ − 5)³。只要对带括号的式子求导，就要先问自己：括号里的导数是什么？
2. Product Rule | 乘积法则
The product rule is needed when two functions are multiplied together, such as x²e³ˣ, x ln x or sin x cos x. There is no way to differentiate these by treating them as a single expression, so the rule must be applied.
当两个函数相乘时，例如 x²e³ˣ、x ln x 或 sin x cos x，就必须使用乘积法则。这类式子无法当作单一表达式直接求导，必须套用公式。
If y = uv, then dy/dx = u dv/dx + v du/dx
Worked example: differentiate y = x²e³ˣ. Take u = x², so du/dx = 2x, and v = e³ˣ, so dv/dx = 3e³ˣ. Then dy/dx = x²(3e³ˣ) + e³ˣ(2x) = x e³ˣ(3x + 2).
例题：求 y = x²e³ˣ 的导数。取 u = x²，则 du/dx = 2x；取 v = e³ˣ，则 dv/dx = 3e³ˣ。于是 dy/dx = x²(3e³ˣ) + e³ˣ(2x) = x e³ˣ(3x + 2)。
+ ln x. This result is used repeatedly in A-Level papers, and it is worth memorising as a pattern.
另一个标准例题是 y = x ln x，其导数 dy/dx = x × (1/x) + ln x × 1 = 1 + ln x。这个结果在 A-Level 试卷中反复出现，值得作为一个固定模式记住。
A useful strategy is to let u be the part that becomes simpler when differentiated, usually a polynomial, and let v be the part that is easy to differentiate or integrate, usually an exponential or trigonometric factor. Always factorise your final answer when the question asks for a simplified form or when you need to find stationary points.
一个实用策略是：令 u 为求导后变得更简单的部分，通常是多项式；令 v 为容易求导或积分的部分，通常是指数或三角函数因子。当题目要求化简或需要求驻点时，一定要对最终答案进行因式分解。
3. Quotient Rule | 商法则
The quotient rule applies when one function is divided by another, for example (2x + 1)/(x − 3), (ln x)/x² or tan x written as sin x / cos x. It is essentially a product rule in disguise, but the standard formula is faster and less error-prone.
当两个函数相除时，例如 (2x + 1)/(x − 3)、(ln x)/x²，或把 tan x 写成 sin x / cos x，就要使用商法则。它本质上是乘积法则的变形，但直接套公式通常更快、更不易出错。
If y = u/v, then dy/dx = (v du/dx − u dv/dx) / v²
dy/dx = [(x − 3)(2) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1)/(x − 3)² = −7/(x − 3)².
例题：求 y = (2x + 1)/(x − 3) 的导数。此处 u = 2x + 1，du/dx = 2；v = x − 3，dv/dx = 1。代入得 dy/dx = [(x − 3)(2) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1)/(x − 3)² = −7/(x − 3)²。
The classic mnemonic is 'low d high minus high d low, over low squared', where 'low' is the denominator v and 'high' is the numerator u. The order matters: reversing the subtraction gives the wrong sign and loses method marks immediately.
经典口诀是'分母乘分子导，减去分子乘分母导，再除以分母的平方'，其中'分母'对应 v，'分子'对应 u。顺序非常重要：把减法顺序颠倒会导致符号错误，直接丢掉步骤分。
Differentiating tan x from sin x / cos x with the quotient rule is a good check of the method, and it produces sec²x, a result you should be able to quote instantly.
用商法则对 tan x = sin x / cos x 求导，是检验该方法的良好练习，结果为 sec²x，这个结论应当能够脱口而出。
4. Differentiating Exponential and Logarithmic Functions | 指数与对数函数求导
Exponential and logarithmic derivatives are heavily examined in Edexcel papers, especially in modelling questions about growth and decay, and in integration questions where ln appears as an answer.
指数与对数函数的导数在 Edexcel 试卷中考查频率很高，尤其是在增长与衰减的建模题，以及答案中会出现 ln 的积分题中。
d/dx (eˣ) = eˣ d/dx (ekx) = k ekx d/dx (ln x) = 1/x d/dx (aˣ) = aˣ ln a
Combined with the chain rule, the general results are d/dx (ef(x)) = f'(x) ef(x) and d/dx (ln f(x)) = f'(x) / f(x). For example, d/dx (e3x²) = 6x e3x² and d/dx (ln(5x + 2)) = 5/(5x + 2).
与链式法则结合后，一般结论为 d/dx (ef(x)) = f'(x) ef(x)，d/dx (ln f(x)) = f'(x) / f(x)。例如 d/dx (e3x²) = 6x e3x²，d/dx (ln(5x + 2)) = 5/(5x + 2)。
The corresponding integrals are equally important: ∫ eˣ dx = eˣ + c, ∫ ekx dx = (1/k) ekx + c, and ∫ (1/x) dx = ln|x| + c. The modulus signs matter, because 1/x is defined for negative x as well.
与之对应的积分同样重要：∫ eˣ dx = eˣ + c，∫ ekx dx = (1/k) ekx + c，∫ (1/x) dx = ln|x| + c。绝对值符号不可省略，因为 1/x 对负的 x 也有定义。
1/x is only valid for x > 0; the modulus form covers the rest. Second, do not confuse ex² with (eˣ)². The first needs the chain rule, the second equals e2x and differentiates to 2e2x.
要警惕两个陷阱。第一，d/dx (ln x) = 1/x 只在 x > 0 时成立，加上绝对值形式才能覆盖其余情况。第二，不要把 ex² 与 (eˣ)² 混淆：前者需要用链式法则，后者等于 e2x，导数为 2e2x。
5. Differentiating Trigonometric Functions | 三角函数求导
Trigonometric derivatives must be learned accurately, because a single sign error can destroy a whole question. All of the standard results below assume the angle is measured in radians, which is the default setting for Edexcel calculus questions.
三角函数导数必须准确记忆，因为一个符号错误就可能毁掉整道题。下面所有标准结果都假设角度以弧度为单位，这也是 Edexcel 微积分题目的默认设定。
| Function y | Derivative dy/dx | Chain rule form |
| sin x | cos x | d/dx sin(ax + b) = a cos(ax + b) |
| cos x | −sin x | d/dx cos(ax + b) = −a sin(ax + b) |
| tan x | sec²x | d/dx tan(ax + b) = a sec²(ax + b) |
| sec x | sec x tan x | d/dx sec(ax + b) = a sec(ax + b) tan(ax + b) |
| cosec x | −cosec x cot x | d/dx cosec(ax + b) = −a cosec(ax + b) cot(ax + b) |
| cot x | −cosec²x | d/dx cot(ax + b) = −a cosec²(ax + b) |
Notice that every derivative of a 'co-' function carries a negative sign. This pattern is a reliable memory aid: the co-functions are cosec, cot and (as a derivative) the derivative of cos.
注意，所有以 co- 开头的函数的导数都带负号。这个规律是可靠的记忆辅助：cosec、cot，以及作为导数的 cos 的导数，都带负号。
r calculator is in degree mode, all of these results become invalid, so always check the display before starting a calculus question.
例题：求 y = sin(3x²) 的导数。使用链式法则，dy/dx = cos(3x²) × 6x = 6x cos(3x²)。如果计算器处于角度制（degree）模式，上述结果全部失效，因此在开始微积分题目前务必检查屏幕显示。
6. Implicit and Parametric Differentiation | 隐函数与参数方程求导
x² + y² = 25, while parametric differentiation handles curves given as x = f(t), y = g(t).
Edexcel 经常考查并非以 y = f(x) 形式写出的曲线。隐函数求导处理 x² + y² = 25 这类方程，而参数方程求导处理 x = f(t)、y = g(t) 形式给出的曲线。
le. For example, differentiating x² + y² = 25 gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. Note how the derivative still contains both x and y, which is normal.
隐函数求导时，对每一项关于 x 求导。任何含 y 的项都必须乘上 dy/dx，这是链式法则的要求。例如对 x² + y² = 25 求导得 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。注意导数中同时含有 x 和 y，这是正常的。
dy/dx = (dy/dt) ÷ (dx/dt)
/dt = 2t and dy/dt = 3t² − 3, giving dy/dx = (3t² − 3) / (2t). The tangent can then be found at any given value of t.
对参数曲线，只要 dx/dt 不为零，梯度就等于 dy/dt 除以 dx/dt。典型题目给出 x = t²、y = t³ − 3t，于是 dx/dt = 2t，dy/dt = 3t² − 3，得 dy/dx = (3t² − 3) / (2t)。之后就可以求出任意给定 t 值处的切线。
/dt). This is required for determining the nature of stationary points on a parametric curve, and it is a common source of lost marks when the final division by dx/dt is forgotten.
参数形式的二阶导数为 d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)。判断参数曲线上驻点的性质时需要用到它，而考生常因忘记最后除以 dx/dt 而失分。
7. Reverse Chain Rule and Integration by Substitution | 反向链式法则与换元积分
Integration is the reverse of differentiation, and Edexcel rewards students who can recognise a derivative hiding inside an integrand. These 'reverse chain rule' patterns solve a large proportion of integration questions quickly.
积分是微分逆运算，Edexcel 非常青睐那些能够识别出被积函数中'隐藏导数'的学生。这些'反向链式法则'模式可以快速解决很大一部分积分题。
- ∫ f'(x) [f(x)]ⁿ dx = [f(x)]n+1 / (n+1) + c, provided n ≠ −1.
- ∫ f'(x) / f(x) dx = ln|f(x)| + c.
- ∫ f'(x) ef(x) dx = ef(x) + c.
- ∫ f'(x) cos f(x) dx = sin f(x) + c.
x = ln(x² + 4) + c, since x² + 4 is always positive and the modulus signs are unnecessary.
例如 ∫ 2x(x² + 1)⁵ dx = (x² + 1)⁶ / 6 + c，因为 x² + 1 的导数 2x 恰好作为因子出现。类似地，∫ (2x)/(x² + 4) dx = ln(x² + 4) + c，因为 x² + 4 恒为正，绝对值符号可以省略。
When the pattern is not obvious, use the substitution method: choose u, find du/dx, rewrite dx as du ÷ (du/dx), integrate in terms of u, and convert back. For definite integrals, either change the limits to u-values or substitute back before evaluating. Changing the limits is usually faster and avoids arithmetic slips.
当模式不明显时，改用换元法：选取 u，求出 du/dx，把 dx 改写为 du ÷ (du/dx)，关于 u 积分后再换回原变量。对于定积分，可以把积分上下限改为 u 的值，也可以先换回原变量再代入。通常改变上下限更快，也能避免计算失误。
8. Integration by Parts | 分部积分
Integration by parts is the counterpart of the product rule and is used for products such as x eˣ, x² sin x, ln x and x ln x. It converts one integral into a simpler one.
分部积分是乘积法则的逆用，用于处理 x eˣ、x² sin x、ln x、x ln x 这类乘积形式。它把一个积分转化为另一个更简单的积分。
∫ u (dv/d
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