📚 Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分
Differentiation is one of the most important tools in Edexcel A-Level Mathematics, linking Pure, Statistics and Mechanics. It allows you to calculate gradients, rates of change and turning points, and it is examined in many forms across all papers. This guide covers every differentiation skill you need: first principles, the power rule, the chain, product and quotient rules, tangents and normals, stationary points, exponential and trigonometric derivatives, and the modelling questions that carry the highest marks.
微分是 Edexcel A-Level 数学中最重要的工具之一，贯穿纯数学、统计与力学。它可用于计算梯度、变化率和驻点，并以多种形式出现在各试卷中。本文覆盖你所需的全部微分技能：第一原理、幂函数法则、链式法则、乘积法则与商法则、切线与法线、驻点、指数函数与三角函数的导数，以及占分最高的建模应用类问题。
1. What Differentiation Actually Means | 微分的真正含义
Differentiation measures how quickly one quantity changes with respect to another. On the curve y = f(x), the derivative at a point is the gradient of the tangent to the curve at that point. A chord drawn between two nearby points gives an average rate of change; as the gap between those points shrinks towards zero, the chord becomes the tangent and the average rate becomes the instantaneous rate.
微分衡量的是一量相对于另一量的变化快慢。在曲线 y = f(x) 上，某点的导数值就是该点处切线的斜率。连接两个邻近点的弦给出平均变化率；当两点间距趋于零时，弦变为切线，平均变化率变为瞬时变化率。
Because the gradient of a curve is different at every point, differentiation produces a new function called the gradient function. There are several standard notations, and Edexcel questions switch freely between them, so you must be comfortable reading all of them.
由于曲线在不同点处斜率不同，求导会产生一个新函数，称为导函数（梯度函数）。导数有多种标准记号，Edexcel 试题会在它们之间自由切换，因此你必须能熟练识别全部记号。
dy/dx = limΔx→0 (Δy / Δx)
- dy/dx — Leibniz notation, useful when variables have physical meaning. | 莱布尼茨记号，变量有物理意义时最方便。
- f '(x) — Lagrange notation, useful for function composition. | 拉格朗日记号，处理复合函数时方便。
- d²y/dx² or f ''(x) — the second derivative, which describes how the gradient itself is changing. | 二阶导数，描述梯度本身如何变化。
2. First Principles and the Derivative Definition | 第一原理与导数定义
The formal definition of the derivative uses the limit of a chord gradient as h tends to 0. This is the definition printed in the formula booklet, and questions that say 'differentiate from first principles' must be answered this way - you will lose all the method marks if you simply apply the power rule.
导数的正式定义使用当 h 趋于 0 时弦梯度的极限。这就是公式册中给出的定义；凡是题干要求 "differentiate from first principles"（从第一原理求导）的题目，都必须用这种方法作答——若直接套用幂函数法则，将失去全部方法分。
f '(x) = limh→0 [ f(x + h) − f(x) ] / h
The three-step routine is always the same. First substitute x + h into the function and expand. Second simplify the difference quotient by cancelling the terms that do not contain h and dividing every remaining term by h. Third let h tend to 0, which removes every term still containing h.
三步流程永远相同。第一步把 x + h 代入函数并展开；第二步化简差商，消去不含 h 的项，并把余下各项除以 h；第三步令 h 趋于 0，此时所有仍含 h 的项都消失。
For example, if f(x) = x², then f(x + h) = (x + h)² = x² + 2xh + h². Subtracting f(x) leaves 2xh + h², and dividing by h gives 2x + h. Letting h → 0 gives f '(x) = 2x.
例如，若 f(x) = x²，则 f(x + h) = (x + h)² = x² + 2xh + h²。减去 f(x) 后余下 2xh + h²，再除以 h 得 2x + h。令 h → 0，得 f '(x) = 2x。
The same routine works for f(x) = x³, giving 3x², and for f(x) = 1/x, giving −1/x². The algebra is longer, but the structure never changes.
同样的流程也适用于 f(x) = x³（得 3x²）以及 f(x) = 1/x（得 −1/x²）。代数过程更长，但结构始终不变。
3. The Power Rule and Basic Polynomials | 幂函数法则与多项式
For any real power n, the power rule states that you bring the power down as a multiplier and reduce the power by one. This is the fastest way to differentiate polynomials and simple powers of x, and it is the single most used rule in the whole specification.
对任意实数幂 n，幂函数法则要求把指数降下来作为系数，再把指数减一。这是对多项式和 x 的简单幂求导最快的方法，也是整个考纲中使用频率最高的法则。
d/dx (xⁿ) = n xⁿ⁻¹
If f(x) = axⁿ where a is a constant, then f '(x) = naxⁿ⁻¹. Constant terms such as −7 differentiate to zero, because a horizontal line has zero gradient. A linear term such as 2x differentiates to 2, because the gradient of a straight line is constant.
若 f(x) = axⁿ，其中 a 为常数，则 f '(x) = naxⁿ⁻¹。像 −7 这样的常数项导数为零，因为水平直线斜率为零；像 2x 这样的一次项导数为 2，因为直线的斜率是常数。
For example, if y = 3x⁵ − 4x³ + 2x − 7, then dy/dx = 15x⁴ − 12x . Check each term separately: 3 × 5 = 15 and 5 − 1 = 4; −4 × 3 = −12 and 3 − 1 = 2; then 2x becomes 2 and −7 becomes 0.
例如，若 y = 3x⁵ − 4x³ + 2x − 7，则 dy/dx = 15x⁴ − 12x² + 2。逐项检查：3 × 5 = 15，指数 5 − 1 = 4；−4 × 3 = −12，指数 3 − 1 = 2；2x 变为 2，−7 变为 0。
When the answer is written with the power zero or negative, Edexcel mark schemes accept either index form or fraction form, but you should normally rewrite negative indices as fractions to match the original question style.
当答案出现零指数或负指数时，Edexcel 评分标准同时接受指数形式与分数形式，但通常应把负指数改写为分数，以与题干风格保持一致。
4. Negative, Fractional and Surd Powers | 负指数、分数指数与根式
Before differentiating any expression containing a root or a fraction, rewrite it as a power of x. This single habit prevents most errors in this topic, because the power rule only ever applies to expressions of the form axⁿ.
在求导任何含根号或分数的表达式之前，先把它改写为 x 的幂。这个习惯能避免本专题中绝大多数错误，因为幂函数法则只适用于 axⁿ 这种形式。
- 1/x = x⁻¹ | 倒数写为负一次幂
- 1/x³ = x⁻³ | 分母三次方对应负三次幂
- √x = x^½ | 平方根即二分之一次幂
- ∛x = x^⅓ | 立方根即三分之一次幂
- 1/√x = x^(−½) | 根号在分母即负二分之一次幂
For example, d/dx (1/x) = d/dx (x⁻¹) = −x⁻² = −1/x². Similarly, if y = 4√x = 4x^½, then dy/dx = 4 × ½ x^(−½) = 2x^(−½) = 2/√x.
例如，d/dx (1/x) = d/dx (x⁻¹) = −x⁻² = −1/x²。同理，若 y = 4√x = 4x^½，则 dy/dx = 4 × ½ x^(−½) = 2x^(−½) = 2/√x。
A harder example: if y = (2x³ + 5) / x, first split the fraction into 2x² + 5x⁻¹, then differentiate term by term to get 4x − 5x⁻². Never apply the quotient rule when a simple division will do - it wastes time and invites mistakes.
一个更难的例子：若 y = (2x³ + 5) / x，先把分数拆为 2x² + 5x⁻¹，再逐项求导得 4x − 5x⁻²。能用简单除法解决时绝不要用商法则——它既浪费时间又容易出错。
5. Sum, Difference and Constant Multiple Rules | 和差法则与常数倍法则
Differentiation is linear, which means you can differentiate a sum or difference term by term, and a constant multiplier can be pulled outside the derivative operator. These two rules, combined with the power rule, handle every polynomial question in the course.
微分是线性的，这意味着你可以逐项求导求和或求差，常数系数也可以提到求导符号之外。这两条法则与幂函数法则结合，就能解决课程中所有多项式题目。
d/dx [ af(x) ± bg(x) ] = af '(x) ± bg '(x)
What you must never do is split a product or a quotient in the same way. The derivative of f(x)g(x) is not f '(x)g '(x), and the derivative of f(x)/g(x) is not f '(x)/g '(x). This single misconception accounts for a large number of lost marks.
但绝不能以同样方式拆开乘积或商。 f(x)g(x) 的导数不是 f '(x)g '(x)，f(x)/g(x) 的导数也不是 f '(x)/g '(x)。这一个误解就会导致大量失分。
| Function | 函数 | Derivative | 导数 |
| 5x⁴ | 20x³ |
| −3x² | −6x |
| 7x | 7 |
| 12 | 0 |
| 6x^½ | 3x^(−½) |
6. The Chain Rule | 链式法则
The chain rule is used when one function is inside another - a 'function of a function'. If y = f(u) and u = g(x), then the rate of change of y with respect to x is the product of the two linked rates.
当函数套函数（"函数的函数"）时使用链式法则。若 y = f(u) 且 u = g(x)，则 y 关于 x 的变化率等于两个相连变化率的乘积。
dy/dx = dy/du × du/dx
In function notation, if y = f(g(x)), then dy/dx = f '(g(x)) × g '(x). This is often remembered as 'differentiate the outside, keep the inside, then multiply by the derivative of the inside'.
用函数记号表示，若 y = f(g(x))，则 dy/dx = f '(g(x)) × g '(x)。这常被记为 "先对外层求导，保留内层，再乘以内层的导数"。
Worked example: differentiate y = (3x + 2)⁵. Take u = 3x + 2 and y = u⁵. Then dy/du = 5u⁴ and du/dx = 3, so dy/dx = 15(3x + 2)⁴. With practice you can do this in one line without writing u down.
例题：求 y = (3x + 2)⁵ 的导数。取 u = 3x + 2，y = u⁵，则 dy/du = 5u⁴，du/dx = 3，故 dy/dx = 15(3x + 2)⁴。熟练后可不必写出 u，一行完成。
Another standard case: y = √x² + 1. Rewrite as y = (x² + 1)^½, so dy/dx = ½(x² + 1)−½ × 2x = x / √x² + 1. Notice how the inner derivative 2x is essential - forgetting it is the most common chain rule error.
另一个标准情形：y = √x² + 1。改写为 y = (x² + 1)^½，则 dy/dx = ½(x² + 1)−½ × 2x = x / √x² + 1。注意内层导数 2x 不可省略——忘记它是链式法则最常见的错误。
7. The Product Rule | 乘积法则
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