📚 Edexcel A Level Maths: Differentiation Rules, Applications and Exam Techniques | 爱德思A Level数学：微分法则、应用与应试技巧
Differentiation is one of the most heavily examined topics in Edexcel A Level Mathematics. It appears throughout the Pure Mathematics papers, it supports mechanics problems such as velocity and acceleration, and it underpins many real-world optimisation questions. This revision guide brings together the key rules, worked examples and exam-focused advice you need, from first principles all the way to implicit and parametric differentiation.
微分是爱德思 A Level 数学中考查最频繁的主题之一。它贯穿纯数学各套试卷，支撑力学中的速度与加速度问题，也是许多现实优化题的数学基础。本篇复习指南汇集了从第一性原理一直延伸到隐函数与参数方程微分的核心法则、典型示例与应试建议。
1. The Gradient Function and First Principles | 梯度函数与第一性原理
The derivative f' us rate of change of y with respect to x at any point where the function is smooth. In Leibniz notation this is written dy/dx.
导数 f'(x) 是曲线 y = f(x) 的梯度函数。它给出了函数光滑处任意点上 y 关于 x 的瞬时变化率。用莱布尼茨记号写作 dy/dx。
The formal definition from first principles is:
第一性原理的形式化定义如下：
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
Edexcel often asks candidates to use this limit definition to differentiate simple functions such as x², x³ or 1/x. Expanding and simplifying fully before letting h tend to zero is essential, and a clear line of algebra is usually worth method marks even if the final answer is wrong.
爱德思考试常要求考生使用该极限定义对 x²、x³ 或 1/x 等简单函数求导。在令 h 趋近于 0 之前，必须先完整展开并化简；即使最终答案有误，清晰的代数过程通常也能拿到方法分。
h². Dividing by h gives 2x + h, and as h approaches zero the expression tends to 2x. So f'(x) = 2x.
例如，若 f(x) = x²，则 f(x + h) − f(x) = (x + h)² − x² = 2xh + h²。除以 h 后得到 2x + h，当 h 趋近于 0 时该表达式趋近于 2x，所以 f'(x) = 2x。
1/x = −h / [x(x + h)]. Dividing by h gives −1 / [x(x + h)], which tends to −1/x² as h → 0.
对于 f(x) = 1/x，同一方法需要先通分：f(x + h) − f(x) = 1/(x + h) − 1/x = −h / [x(x + h)]。除以 h 得 −1 / [x(x + h)]，当 h → 0 时趋近于 −1/x²。
2. Standard Derivatives and the Power Rule | 标准导数与幂法则
For any real power n, the power rule states that if f(x) = xⁿ then f' ⁻¹. This single rule is the foundation for polynomials, roots and many rational functions.
对任意实数次幂 n，幂法则指出：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。这一条法则就是多项式、根式以及许多有理函数求导的基础。
Before differentiating, rewrite every term in index form. Terms such as √x, 1/x² and 3/x³ become x1/2, x⁻² and 3x⁻³, which can then be handled by the power rule directly.
求导之前，先把每一项改写成指数形式。像 √x、1/x²、3/x³ 这样的项应改写为 x1/2、x⁻²、3x⁻³，然后直接套用幂法则。
Key standard derivatives are summarised below. All of them should be memorised, because Edexcel does not supply them in the A Level Mathematics formula booklet.
主要标准导数总结如下。这些都必须背下来，因为爱德思 A Level 数学的公式手册里并不提供它们。
| f(x) | f'(x) | 中文说明 |
|---|---|---|
| k (constant) | 0 | 常数的导数为零 |
| xⁿ | nxⁿ⁻¹ | 幂法则，适用于负数与分数指数 |
| eˣ | eˣ | 指数函数求导后不变 |
| eᵏˣ | k eᵏˣ | 线性复合，常数因子提到前面 |
| ln x | 1/x | 定义域为 x > 0 |
| sin x | cos x | 角度必须用弧度制 |
| cos x | −sin x | 注意负号，最易失分 |
| sin kx | k cos kx | 链式法则的简单形式 |
| cos kx | −k sin kx | 负号与系数都要保留 |
of the derivatives, and constant factors stay in place. So d/dx (4x³ − 5x + 7) = 12x² − 5.
微分是线性的：和的导数等于导数的和，常数因子可以保留不动。因此 d/dx (4x³ − 5x + 7) = 12x² − 5。
3. Tangents, Normals and the Equation of a Line | 切线与法线
The value of dy/dx at a point is the gradient of the tangent to the curve at that point. This is the single most common application of differentiation in the Pure papers.
在某一点处 dy/dx 的取值，就是曲线在该点切线的斜率。这是纯数学试卷中最常见的微分应用。
The standard method is short and should be automatic:
标准步骤很短，应当做到自动化：
- Differentiate the curve to obtain dy/dx.
- Substitute the x-coordinate of the point to find the numerical gradient m.
- Substitute the point and m into y − y₁ = m(x − x₁).
- Rearrange into the form the question asks for, usually y = mx + c or ax + by + c = 0.
- 先把曲线求导得到 dy/dx。
- 代入该点的横坐标，求出数值斜率 m。
- 把该点与 m 代入 y − y₁ = m(x − x₁)。
- 整理成题目要求的形式，通常是 y = mx + c 或 ax + by + c = 0。
The normal is perpendicular to the tangent, so its gradient is −1/m. Remember that a horizontal tangent has gradient 0 and a vertical tangent has an undefined gradient, in which case the normal is horizontal.
法线与切线垂直，因此其斜率为 −1/m。注意水平切线的斜率为 0，竖直切线的斜率不存在，此时法线是水平的。
2), that is y = 9x − 16. The normal has gradient −1/9 and equation y − 2 = −(1/9)(x − 2).
示例：对 y = x³ − 3x 在点 (2, 2) 处，dy/dx = 3x² − 3，故 m = 9。切线为 y − 2 = 9(x − 2)，即 y = 9x − 16。法线斜率为 −1/9，方程为 y − 2 = −(1/9)(x − 2)。
4. The Second Derivative and Stationary Points | 二阶导数与驻点
Differentiating a second time gives f''(x), written d²y/dx². It measures the rate of change of the gradient, which tells you how the curve bends.
再求导一次得到 f''(x)，记作 d²y/dx²。它衡量斜率的変化率，也就是曲线弯曲的方式。
= 0. There are three common types, and the second derivative test classifies them quickly:
驻点出现在 dy/dx = 0 处。常见有三种类型，用二阶导数判别可以快速分类：
- If f''(x) < 0, the point is a local maximum.
- If f''(x) > 0, the point is a local minimum.
- If f''(x) = 0, the test is inconclusive; use a table of signs of f'(x) on either side.
- 若 f''(x) < 0，该点为局部极大值点。
- 若 f''(x) > 0，该点为局部极小值点。
- 若 f''(x) = 0，判别法失效，需用 f'(x) 两侧的符号表来判断。
A point where the curve changes concavity is called a point of inflection. It is not always stationary, and examiners often include one in a diagram to check that you do not assume every turning point is a maximum or a minimum.
曲线凹凸性发生改变的点称为拐点。拐点不一定是驻点；考官常在图示中安排一个拐点，检验你是否会误以为每个转折点都是极大或极小。
5. The Chain Rule | 链式法则
The chain rule is used for composite functions, where one function sits inside another. In Leibniz notation:
链式法则用于复合函数，也就是一个函数套在另一个函数里面。用莱布尼茨记号写作：
dy/dx = (dy/du) × (du/dx)
In words: differentiate the outside function, keep the inside unchanged, then multiply by the derivative of the inside function.
用文字表述：先对外层函数求导并保持内层不变，再乘以内层函数的导数。
Typical examples include y = (3x + 2)⁵, y = e4x − 1, y = sin(2x) and y = ln(5x + 3). The answers are 15(3x + 2)⁴, 4e4x − 1, 2cos(2x) and 5/(5x + 3) respectively.
典型例子包括 y = (3x + 2)⁵、y = e4x − 1、y = sin(2x) 与 y = ln(5x + 3)。它们的导数分别是 15(3x + 2)⁴、4e4x − 1、2cos(2x) 与 5/(5x + 3)。
When the inner function is itself a power, rewrite before differentiating. For y = √x² + 1 = (x² + 1)1/2, the derivative is (1/2)(x² + 1)−1/2 × 2x = x / √x² + 1.
当内层函数本身也是幂时，先改写再求导。对 y = √x² + 1 = (x² + 1)1/2，导数为 (1/2)(x² + 1)−1/2 × 2x = x / √x² + 1。
6. The Product Rule and the Quotient Rule | 乘积法则与商法则
Use the product rule when two functions are multiplied together. If y = uv, then:
当两个函数相乘时使用乘积法则。若 y = uv，则：
dy/dx = u (dv/dx) + v (du/dx)
Use the quotient rule when one function is divided by another. If y = u / v, then:
当一个函数除以另一个函数时使用商法则。若 y = u / v，则：
dy/dx = [v (du/dx) − u (dv/dx)] / v²
In the quotient rule the numerator order matters. A common mnemonic is "low d high minus high d low, over low squared", where "low" is the denominator and "high" is the numerator.
商法则中分子的顺序很关键。常用口诀是"分母乘分子的导数，减去分子乘分母的导数，再除以分母的平方"。
Worked examples: for y = x²eˣ, take u = x² and v = eˣ, giving dy/dx = 2xeˣ + x²eˣ = xeˣ(2 + x). For y = ln x / x, take u = ln x and v = x, giving dy/dx = (1 − ln x)/x².
示例：对 y = x²eˣ，取 u = x²、v = eˣ，得 dy/dx = 2xeˣ + x²eˣ = xeˣ(2 + x)。对 y = ln x / x，取 u = ln x、v = x，得 dy/dx = (1 − ln x)/x²。
Always simplify and factorise your answer. Edexcel mark schemes frequently award the final mark only for a correctly factorised form, and simplified answers make follow-up parts such as finding stationary points far easier.
务必化简并因式分解答案。爱德思评分标准常常只给正确因式分解形式最后那一分，而化简后的答案也让后续求驻点等小题容易得多。
7. Exponential and Logarithmic Functions | 指数函数与对数函数
The exponential function is its own derivative: d/dx (eˣ) = eˣ. Combined with the chain rule, d/dx (eᵏˣ) = k eᵏˣ, and more generally d/dx (eu) = eu × (du/dx).
指数函数的导数就是它本身：d/dx (eˣ) = eˣ。结合链式法则，d/dx (eᵏˣ) = k eᵏˣ，更一般地 d/dx (eu) = eu × (du/dx)。
minder that multiplying by a constant inside a logarithm only shifts the graph vertically.
对自然对数，当 x > 0 时 d/dx (ln x) = 1/x。配合链式法则，d/dx (ln(kx)) = 1/x，这也提醒我们：对数内部乘常数只是把图像上下平移。
These two results combine beautifully with the product and quotient rules. For y = e2x ln x, the product rule gives dy/dx = 2e2x ln x + e2x/x, which factorises to e2x(2 ln x + 1/x).
这两个结果与乘积法则、商法则配合得非常好。对 y = e2x ln x，乘积法则给出 dy/dx = 2e2x ln x + e2x/x，因式分解为 e2x(2 ln x + 1/x)。
Exam tip: if the question says "show that" and gives a target expression, work from one side only and keep every line visible, since unsupported jumps lose accuracy marks.
应试提示：若题目说"证明"，并给出了目标表达式，就要只从一边开始推导并写出每一行，因为跳跃过大的步骤会丢掉准确分。
8. Trigonometric Derivatives | 三角函数的导数
negative sign in the second one, which is the most frequently dropped mark in this topic.
两个核心结果是 d/dx (sin x) = cos x 与 d/dx (cos x) = −sin x。注意第二个结果中的负号，这是本主题中最常被漏掉的一分。
These results only hold when the angle is measured in radians. If a question involves degrees, convert to radians first, or expect the constants to change.
这些结果只在角度以弧度为单位时成立。若题目涉及角度制，应先换算为弧度，否则常数会发生变化。
d/dx (cos kx) = −k sin kx. For example, d/dx (3 sin 2x − 4 cos 5x) = 6 cos 2x + 20 sin 5x.
结合链式法则，标准推广式为 d/dx (sin kx) = k cos kx 与 d/dx (cos kx) = −k sin kx。例如 d/dx (3 sin 2x − 4 cos 5x) = 6 cos 2x + 20 sin 5x。
Trigonometric derivatives also appear in modelling questions, where you may be asked for the maximum gradient of a curve or the first time a gradient reaches a particular value. Setting the derivative equal to a constant and solving a trigonometric equation is the usual route.
三角函数的导数也出现在建模题中，可能要求某曲线斜率的最大值，或斜率首次取到某个值的时刻。通常的做法是把导数设为常数，然后解一个三角方程。
9. Implicit Differentiation | 隐函数求导
Differentiate every term with respect to x, remembering that y is itself a function of x.
当 y 没有显式表示为 x 的函数时，例如 x² + y² = 25 或 x³ + xy = 2y²，就要用隐函数求导。对每一项关于 x 求导，并始终记住 y 本身是 x 的函数。
y picks up a factor of dy/dx. So d/dx (y²) = 2y (dy/dx), and d/dx (sin y) = cos y (dy/dx).
关键一步是：关于 y 的函数求导时会带出因子 dy/dx。因此 d/dx (y²) = 2y (dy/dx)，d/dx (sin y) = cos y (dy/dx)。
ies in full: d/dx (xy) = y + x (dy/dx). This is the step where most candidates lose marks.
当出现 x 与 y 的乘积时，必须完整使用乘积法则：d/dx (xy) = y + x (dy/dx)。这一步是大多数考生失分的地方。
esult is 2x + 2y (dy/dx) = 0, so dy/dx = −x/y. A tangent at (3, 4) then has gradient −3/4.
求导之后，把所有含 dy/dx 的项集中到一边，提取公因式 dy/dx 再作除法。对 x² + y² = 25，得到 2x + 2y (dy/dx) = 0，故 dy/dx = −x/y。在点 (3, 4) 处的切线斜率为 −3/4。
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