📚 Differentiation Techniques for Edexcel A-Level Maths | Edexcel A-Level 数学微分技巧
This revision guide covers the core differentiation techniques required for Edexcel A-Level Mathematics, including first principles, the power rule, the chain, product and quotient rules, trigonometric and exponential derivatives, implicit and parametric differentiation, and the exam-focused skills that turn method marks into full marks. Every technique is paired with the way it is actually assessed, plus the mistakes that cost students the most marks.
本复习指南涵盖 Edexcel A-Level 数学所需的核心微分技巧，包括第一原理、幂法则、链式法则、乘积法则与商法则、三角函数与指数函数求导、隐函数与参数方程求导，以及能把"方法分"变成满分的考试技巧。每个技巧都会配上它在真题中的实际考法，以及最容易失分的常见错误。
1. First Principles and the Gradient Function | 第一原理与梯度函数
The derivative f' t, and it is defined as the limit of the gradient of a chord joining two nearby points on the curve.
导数 f'(x) 衡量函数 f(x) 的瞬时变化率。从几何上看，它表示曲线 y = f(x) 在某点处切线的斜率，其定义是曲线上两个相邻点所连弦的斜率在极限意义下的取值。
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
oth notations appear in Edexcel questions, so you should be comfortable reading and writing each of them.
等价的写法是用 x 与 y 的微小增量表示：弦的斜率为 δy/δx，而导数是当 δx → 0 时该比值趋近的值。Edexcel 题目中两种记号都会出现，因此你需要能熟练地读写它们。
dy/dx = lim(δx→0) δy/δx
btracting f(x) leaves 2xh + h², and dividing by h gives 2x + h. Letting h → 0 then gives f'(x) = 2x.
对于简单多项式，这个极限可以直接算出来。以 f(x) = x² 为例：展开得 f(x + h) = x² + 2xh + h²，减去 f(x) 后剩 2xh + h²，再除以 h 得 2x + h，令 h → 0 即得 f'(x) = 2x。
ng h = 0, because examiners award method marks for this structure even if a later algebraic slip occurs.
考试中，你必须清晰展示展开、约去 h 以及取极限的全过程。在代入 h = 0 之前一定要写出极限表达式，因为即使后面出现代数失误，阅卷人也会根据这一结构给出方法分。
The same method works for f(x) = x³, giving f' (x) = 1/x, where the common denominator step is the key algebraic move. Questions that say "from first principles" or "using the definition of the derivative" can never be answered by quoting the power rule.
同样的方法适用于 f(x) = x³，得到 f'(x) = 3x²，也适用于 f(x) = 1/x 这类简单有理函数，其中通分是最关键的代数步骤。凡题目要求"用第一原理"或"利用导数定义"，都不能直接引用幂法则作答。
2. The Power Rule and Basic Derivatives | 幂法则与基本导数
For any real constant n, the derivative of xⁿ is n xⁿ⁻¹. This single rule underpins almost every polynomial, rational and root-based differentiation question in the Edexcel specification.
对于任意实数 n，xⁿ 的导数为 n xⁿ⁻¹。这一条法则支撑了 Edexcel 考纲中几乎所有的多项式、有理式和根式求导题。
d/dx (xⁿ) = n xⁿ⁻¹
The rule works equally well for positive integers, negative powers, and fractional powers. Expressions such as √x and 1/x² are not exceptions: they are simply powers that must be rewritten first.
该法则同样适用于正整数、负整数次幂以及分数次幂。像 √x 和 1/x² 这样的表达式并不是例外，它们只是需要先改写成分数指数形式的幂。
√x = x1/2, so its derivative is ½ x−1/2, which is usually rewritten as 1 / (2√x). Likewise 1/x² = x⁻², whose derivative is −2x⁻³, or −2/x³.
√x = x1/2，其导数为 ½ x−1/2，通常改写为 1 / (2√x)。同理 1/x² = x⁻²，导数为 −2x⁻³，即 −2/x³。
Any constant term differentiates to zero, and the derivative of x itself is 1. These two facts are the most frequently forgotten details in otherwise correct solutions, so build them into your routine.
任何常数项求导都得 0，而 x 本身的导数为 1。这两个事实是正确解答中最常被漏掉的细节，应把它们变成固定习惯。
A useful exam habit is to convert the entire expression into index form before differentiating, then convert back into surd or fraction form at the end. Marks are usually awarded for a correctly simplified derivative, not for a half-converted expression.
一个实用的考试习惯是：求导前先把整个式子化成分数指数形式，最后再转回根式或分数形式。分数通常给的是化简正确的导数，而不是半转换的式子。
3. Polynomials, Sums and Constant Multiples | 多项式、和与常数倍
Differentiation is linear, which means the derivative of a sum is the sum of the derivatives, and a constant multiplier can be taken outside the derivative operator. This lets you differentiate any polynomial term by term.
微分是线性的：和的导数等于导数的和，常数因子可以直接提到求导符号外面。因此你可以逐项求导任何多项式。
d/dx [a f(x) + b g(x)] = a f'(x) + b g'(x)
dy/dx = 12x³ − 10x + 7. Each term is handled independently and the constant −2 disappears.
例如，若 y = 3x⁴ − 5x² + 7x − 2，则 dy/dx = 12x³ − 10x + 7。每一项独立处理，常数 −2 求导后消失。
he same linearity applies when the powers are negative or fractional. If y = 4x³ + 2/x − 3√x, rewrite first as y = 4x³ + 2x⁻¹ − 3x1/2, then differentiate to get dy/dx = 12x² − 2x⁻² − (3/2)x−1/2.
当幂次为负数或分数时，线性性质同样成立。若 y = 4x³ + 2/x − 3√x，先改写为 y = 4x³ + 2x⁻¹ − 3x1/2，再求导得 dy/dx = 12x² − 2x⁻² − (3/2)x−1/2。
Typical errors here are dropping the negative sign when the index is negative, and forgetting to reduce the index by one. Checking that each index has decreased by exactly 1 is a fast self-verification in timed conditions.
这里常见的错误是：指数为负时丢掉负号，以及忘记把指数减 1。在限时考试中，快速检查每个指数是否恰好减少了 1，是很好的自我验证方法。
When a question asks for the gradient at a specific point, differentiate first and substitute afterwards. Substituting the x-value into the original function gives the coordinate you may need for a tangent or normal line later in the same question.
当题目要求某点的斜率时，应先求导再代入。把 x 值代入原函数则得到坐标，这个坐标在后面求切线或法线时往往还会用到。
4. The Chain Rule for Composite Functions | 复合函数的链式法则
he derivative of the inside. In Leibniz notation this is written as dy/dx = dy/du × du/dx.
当函数由外层函数和内层函数复合而成时，应先对外层求导，再乘以对内层求导的结果。用莱布尼茨记号可写成 dy/dx = dy/du × du/dx。
d/dx [f(g(x))] = f'(g(x)) × g'(x)
ide gives 5(3x + 2)⁴, and multiplying by the inner derivative 3 gives dy/dx = 15(3x + 2)⁴.
对于 y = (3x + 2)⁵，外层是五次幂，内层是 3x + 2。外层求导得 5(3x + 2)⁴，再乘以内层导数 3，得 dy/dx = 15(3x + 2)⁴。
The rule extends to roots and reciprocals. For y = √2x² + 1, rewrite as (2x² + 1)1/2, differentiate to ½(2x² + 1)−1/2 × 4x, and simplify to 2x / √2x² + 1.
该法则也适用于根式和倒数。对于 y = √2x² + 1，改写为 (2x² + 1)1/2，求导得 ½(2x² + 1)−1/2 × 4x，化简为 2x / √2x² + 1。
rally with trigonometric and exponential functions: d/dx [sin(3x)] = 3cos(3x), and d/dx [ex²] = 2x ex². In every case the multiplication by the inner derivative is compulsory.
它与三角函数、指数函数结合得非常自然：d/dx [sin(3x)] = 3cos(3x)，d/dx [ex²] = 2x ex²。任何情况下，都必须乘以内层导数。
The single most common error in Edexcel papers is differentiating the outside correctly but forgetting the inner derivative, for instance writing 5(3x + 2)⁴ instead of 15(3x + 2)⁴. If your answer looks slightly "too simple", check the inner factor immediately.
Edexcel 试卷中最常见的错误是：外层求导正确但漏掉内层导数，例如把答案写成 5(3x + 2)⁴ 而不是 15(3x + 2)⁴。如果你的答案看起来"太简单了"，请立刻检查内层因子。
5. The Product Rule | 乘积法则
When two functions are multiplied together, you cannot simply multiply their derivatives. The product rule states that the derivative of uv is u times the derivative of v plus v times the derivative of u.
当两个函数相乘时，不能直接把各自的导数相乘。乘积法则指出：uv 的导数等于 u 乘以 v 的导数，加上 v 乘以 u 的导数。
d/dx (uv) = u (dv/dx) + v (du/dx)
For y = x² eˣ, take u = x² and v = eˣ. Then du/dx = 2x and dv/dx = eˣ, so dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2). Factorising the answer is usually expected at A-Level.
对于 y = x² eˣ，取 u = x²、v = eˣ，则 du/dx = 2x、dv/dx = eˣ，所以 dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2)。在 A-Level 中，通常要求把答案因式分解。
x × 1 = 1 + ln x, a result that appears repeatedly in exam questions on maxima and minima.
对于 y = x ln x，取 u = x、v = ln x，则 dy/dx = x × (1/x) + ln x × 1 = 1 + ln x。这个结果在以极值为背景的真题中反复出现。
² cos x + 2x sin x. Choosing u as the polynomial term is usually sensible, because differentiating it produces something simpler and the product becomes easier to tidy up.
对于 y = x² sin x，法则给出 dy/dx = x² cos x + 2x sin x。通常把多项式部分取作 u 比较明智，因为它求导后会变简单，乘积也更容易整理。
The product rule is also a useful alternative to the quotient rule. A fraction such as x / (x + 1) can be rewritten as x(x + 1)⁻¹ and differentiated with the product and chain rules together.
乘积法则也是商法则的有用替代方案。像 x / (x + 1) 这样的分式，可以改写为 x(x + 1)⁻¹，然后结合乘积法则与链式法则求导。
6. The Quotient Rule | 商法则
When one function is divided by another, use the quotient rule. In words: denominator times derivative of numerator, minus numerator times derivative of denominator, all divided by the denominator squared.
当一个函数除以另一个函数时，使用商法则。用文字表述：分母乘以分子的导数，减去分子乘以分母的导数，再除以分母的平方。
d/dx (u/v) = [v (du/dx) − u (dv/dx)] / v²
dx = [(x − 3)(2) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1) / (x − 3)² = −7 / (x − 3)².
对于 y = (2x + 1) / (x − 3)，取 u = 2x + 1、v = x − 3，则 /dx = [(x − 3)(2) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1) / (x − 3)² = −7 / (x − 3)²。
The order of the subtraction matters and cannot be reversed. If the numerator terms are written the other way round, the sign of the whole derivative flips, which is one of the most heavily penalised errors in this topic.
减法的顺序至关重要，不能颠倒。如果把分子的两项写反，整个导数的符号都会改变，这是本专题中被扣分最严重的错误之一。
The quotient rule also gives the derivative of tan x when applied to sin x / cos x, producing sec²x. This derivation is worth remembering because it links the quotient rule directly to a standard trigonometric result.
把商法则用于 sin x / cos x，还能得到 tan x 的导数 sec²x。这个推导值得记住，因为它把商法则与一个标准三角结果直接联系了起来。
Before starting, check whether the quotient simplifies. If the numerator can be divided by the denominator term by term, rewriting the expression as a sum of powers is faster and less error-prone than applying the rule.
动手前先检查分式能否化简。如果分子可以逐项除以分母，那么把式子改写为幂的和，会比直接套用商法则更快、更不容易出错。
7. Trigonometric Derivatives | 三角函数求导
For angles measured in radians, the derivatives of sine, cosine and tangent follow the standard cycle. Radians are essential: these results are not valid when the angle is measured in degrees.
当角度以弧度为单位时，正弦、余弦和正切的导数遵循标准规律。弧度必不可少：若角度以度为单位，这些结论并不成立。
d/dx (sin x) = cos x d/dx (cos x) = −sin x d/dx (tan x) = sec²x
osec²x. These follow from the quotient rule applied to the corresponding reciprocal identities.
在进阶数学中，你还需要掌握 d/dx (sec x) = sec x tan x、d/dx (cosec x) = −cosec x cot x 以及 d/dx (cot x) = −cosec²x。它们都可以由倒数关系配合商法则推出。
sin(3x)] = 3cos(3x), d/dx [cos(x²)] = −2x sin(x²), and d/dx [tan(2x + 1)] = 2sec²(2x + 1).
复合三角函数依赖链式法则。因此 d/dx [sin(3x)] = 3cos(3x)，d/dx [cos(x²)] = −2x sin(x²)，d/dx [tan(2x + 1)] = 2sec²(2x + 1)。
A frequent exam trap is a question that silently assumes degree measure in a modelling context. If the angle is in degrees, the derivative of sin θ picks up an extra factor of π/180, so read the wording of every modelling question carefully.
常见的考试陷阱是：建模题中悄悄使用了角度制。如果角度以度为单位，sin θ 的导数会多出一个 π/180 的因子，因此每一道建模题都要仔细读题。
Remember too that differentiating twice returns you to the original function with a possible sign change: the second derivative of sin x is −sin x, and of cos x is −cos x. This cyclical behaviour is what makes sine and cosine so useful in modelling periodic motion.
还要记住，求导两次会回到原函数本身，只是可能带一个符号：sin x 的二阶导数是 −sin x，cos x 的二阶导数是 −cos x。正是这种循环性质，使正弦和余弦在周期运动的建模中格外有用。
8. Exponential and Logarithmic Derivatives | 指数函数与对数函数求导
The exponential function eˣ is the only function that is its own derivative. This self-reproducing property is why e appears throughout calculus, and it makes exponential derivatives among the easiest to remember.
指数函数 eˣ 是唯一等于自身导数的函数。正是这种"自我复制"的性质让 e 贯穿整个微积分，也使指数函数求导成为最容易记忆的一类。
d/dx (eˣ) = eˣ d/dx (ekx) = k ekx d/dx (aˣ) = aˣ ln a
/x for x > 0. More generally, for any positive differentiable function, d/dx [ln f(x)] = f'(x) / f(x), a form that appears constantly in integration and in optimisation problems.
对于自然对数，当 x > 0 时，d/dx (ln x) = 1/x。更一般地，对任意可导的正函数，d/dx [ln f(x)] = f'(x) / f(x)，这一形式在积分和最优化问题中频繁出现。
Combining with the chain rule covers the majority of exam cases. So d/dx [e3x] = 3e3x, d/dx [ex²] = 2x ex² x)] = 1/x, because the constant factor inside the logarithm disappears on differentiation.
结合链式法则即可覆盖大多数考题：d/dx [e3x] = 3e3x，d/dx [ex²] = 2x ex²，而 d/dx [ln(5x)] = 1/x，因为对数内部的常数因子在求导后会消失。
Watch the distinction between ex² and (eˣ)². The first differentiates to 2x ex², while the second is simply e2x and differentiates to 2e2x. Misreading the position of the exponent leads to a completely wrong answer.
注意区分 ex² 与 (eˣ)²。前者求导得 2x ex²，而后者其实就是 e2x，导数为 2e2x。看错指数位置会导致答案完全错误。
m demand is to find the exact coordinates of a stationary point on a curve such as y = x eˣ lowed by careful solving of the resulting equation, usually by factorising or by using ln.
考试中常要求求出 y = x eˣ 或 y = ln x / x 这类曲线上驻点的精确坐标。两者都需要先使用乘积或商法则，再通过因式分解或取对数来仔细解出方程。
9. Implicit Differentiation | 隐函数求导
o find dy/dx directly from an equation such as x² + y² = 25, without rearranging it first.
有些 x 与 y 的关系并不以 y = f(x) 的形式给出。隐函数求导让你可以直接从 x² + y² = 25 这类方程求出 dy/dx，而不必先变形。
taining y therefore needs the chain rule: d/dx (y²) = 2y dy/dx, and d/dx (y³) = 3y² dy/dx.
对等式两边关于 x 求导，把 y 视为 x
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