Edexcel A-Level Maths: Mastering Differentiation | 爱德思 A-Level 数学：掌握微分法

📚 Edexcel A-Level Maths: Mastering Differentiation | 爱德思 A-Level 数学：掌握微分法

Differentiation is one of the most frequently examined topics in Edexcel A-Level Pure Mathematics. It provides the tools to measure instantaneous rates of change, determine gradients of curves, and solve real-world optimisation problems. A solid command of derivative rules, stationary points, and tangents is essential for high marks across Paper 1 and Paper 2. This guide walks through the definition from first principles, the standard rules for powers, products, quotients, exponentials, logarithms and trigonometric functions, and the applied techniques that examiners reward most consistently.

微分是爱德思 A-Level 纯数学中最常考的专题之一。它提供了度量瞬时变化率、求曲线斜率和解决现实优化问题的工具。扎实掌握求导法则、平稳点与切线，是在 Paper 1 和 Paper 2 中取得高分的关键。本文将从第一原理的定义出发，系统梳理幂函数、乘积、商、指数、对数与三角函数的求导规则，并总结考官最看重的应用题技巧。

1. First Principles and the Definition of Derivative | 导数定义与第一原理

The derivative measures the instantaneous rate of change of a function. It is defined as the limit of the average rate of change over an interval as the width of that interval shrinks towards zero. Geometrically, this is the gradient of a chord PQ tending to the gradient of the tangent at P. For a function f(x), the derivative at a point is written f'(x) and is given by the following limit, provided it exists:

f'(x) = lim(h→0) [f(x + h) − f(x)] ÷ h

导数刻画函数的瞬时变化率，其定义是：当区间宽度不断缩小趋于零时，平均变化率的极限。从几何上看，这就是弦 PQ 的斜率趋于点 P 处切线斜率的过程。对于函数 f(x)，某点处的导数记为 f'(x)，在极限存在的前提下由下式给出：

f'(x) = lim(h→0) [f(x + h) − f(x)] ÷ h

(x) = x², then f(x + h) − f(x) = (x + h)² − x² = 2xh + h², so the difference quotient is 2x + h, which tends to 2x as h → 0. Edexcel regularly awards four or five marks for a first-principles derivation, so every algebraic line must be written down clearly.

用第一原理求导的步骤是：展开 f(x + h)，减去 f(x)，化简分子，除以 h，最后才令 h → 0。例如，若 f(x) = x²，则 f(x + h) − f(x) = (x + h)² − x² = 2xh + h²，于是差商为 2x + h，当 h → 0 时趋于 2x。爱德思考试中对第一原理推导通常给出 4 至 5 分，因此每一步代数过程都必须写清楚。

Several notations describe the same object, and you should be comfortable switching between them under exam pressure.

同一个对象有多种记号，考试中必须能在它们之间自如切换。

2. The Power Rule for Differentiating xⁿ | 幂函数求导法则

For any real number n, the derivative of xⁿ is n xⁿ⁻¹: multiply by the power, then reduce the power by one. This single rule handles polynomials, surds and reciprocals, provided the expression is first rewritten as a power of x. Rewriting before differentiating is the habit that saves the most marks in the whole topic.

对任意实数 n，xⁿ 的导数为 n xⁿ⁻¹：先乘以指数，再把指数减一。只要先把表达式改写为 x 的幂，这一条规则就能处理多项式、根式和倒数。先改写再求导，是整个专题中最能省分的习惯。

f(x) = xⁿ → f'(x) = n xⁿ⁻¹

• x⁵ → 5x⁴
• √x = x^½ → ½ x^−½ = 1 ÷ (2√x)
• 1 ÷ x = x⁻¹ → −x⁻² = −1 ÷ x²
• 3 ÷ x² = 3x⁻² → −6x⁻³ = −6 ÷ x³
• x = x¹ → 1
• k (a constant) → 0

Students are also expected to differentiate small positive integer powers from first principles, since that derivation is the justification for the power rule itself. The algebra for x³ and x⁴ follows the same pattern as x², using the binomial expansion of (x + h)ⁿ.

考纲还要求能用第一原理推导小的正整数次幂，因为这正是幂法则本身的依据。x³ 与 x⁴ 的推导与 x² 完全同构，只是使用 (x + h)ⁿ 的二项展开。

3. Sums, Differences and Constant Multiples | 和差法则与常数倍法则

n be applied term by term. For constants a and b, the derivative of a f(x) ± b g(x) is a f'(x) ± b g' rentiates to zero because its graph is horizontal and its rate of change is therefore nil.

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微分是线性运算，因此可以逐项进行。对于常数 a 与 b，a f(x) ± b g(x) 的导数为 a f'(x) ± b g'(x)。常数项的导数为零，因为它的图像是水平的，变化率自然为零。

d ÷ dx [a f(x) ± b g(x)] = a f'(x) ± b g'(x)

ked example: for f(x) = 3x⁴ − 5x² + 7x − 2, differentiate each term separately to obtain f' first, such as expanding brackets, splitting a fraction, or converting roots into powers.

例题：对 f(x) = 3x⁴ − 5x² + 7x − 2 逐项求导，得到 f'(x) = 12x³ − 10x + 7。大多数考题给出的表达式都需要先化简，例如展开括号、拆分分式，或把根式改写为幂。

• y = x(x² − 4) = x³ − 4x → dy ÷ dx = 3x² − 4
• y = (x² + 3x) ÷ x = x + 3 → dy ÷ dx = 1, valid for x ≠ 0
• y = (2x + 1)² = 4x² + 4x + 1 → dy ÷ dx = 8x + 4

Expanding is usually faster and safer than the product rule when the expression is a simple polynomial product, but it is not always possible, which is why the rules in Sections 6 to 8 remain essential.

当表达式是简单的多项式乘积时，展开通常比用乘积法则更快、更稳妥；但并非所有情况都能展开，这正是第 6 至 8 节规则依然必要的原因。

4. Tangents and Normals | 切线与法线

ngent is written as y − y₁ = m(x − x₁). The normal is the line perpendicular to the tangent at that point, so its gradient is −1 ÷ m, and it satisfies the same point-gradient form.

曲线在某点处切线的斜率，就是导数在该点的取值。已知斜率 m 与点 (x₁, y₁) 后，切线方程为 y − y₁ = m(x − x₁)。法线是过该点且与切线垂直的直线，其斜率为 −1 ÷ m，同样使用点斜式。

Tangent: y − y₁ = m(x − x₁)    Normal: y − y₁ = (−1 ÷ m)(x − x₁)

g the point (3, 2). The tangent is y − 2 = 2(x − 3), that is y = 2x − 4. The normal has gradient −½, so y − 2 = −½(x − 3), which simplifies to y = −½x + 3½.

例题：求 y = x² − 4x + 5 在 x = 3 处的切线与法线。先求 dy ÷ dx = 2x − 4，当 x = 3 时斜率为 2。纵坐标为 9 − 12 + 5 = 2，即点 (3, 2)。切线为 y − 2 = 2(x − 3)，即 y = 2x − 4。法线斜率为 −½，故 y − 2 = −½(x − 3)，化简得 y = −½x + 3½。

• Always state the coordinates of the point before writing an equation | 写方程前先写出点的坐标
• Leave the answer in the form requested, usually y = mx + c or ax + by + c = 0 | 按要求的形式作答，通常是 y = mx + c 或 ax + by + c = 0
• If the tangent is horizontal, the normal is vertical and takes the form x = a | 若切线水平，则法线竖直，形如 x = a

5. Stationary Points and the Second Derivative | 平稳点与二阶导数

. They are classified as local maxima, local minima, or points of inflection. The nature of a stationary point is usually determined by the sign of the second derivative, which is found by differentiating a second time.

平稳点出现在斜率为零处，即 dy ÷ dx = 0 的地方。它们分为局部极大值、局部极小值和拐点。平稳点的性质通常由二阶导数的符号判定，而二阶导数就是对导数再求一次导。

If d²y ÷ dx² > 0, the point is a local minimum; if d²y ÷ dx² < 0, it is a local maximum.

t equals −6, so (0, 4) is a local maximum; at x = 2 it equals 6, so (2, 0) is a local minimum.

例题：对于 y = x³ − 3x² + 4，dy ÷ dx = 3x² − 6x = 3x(x − 2)，故平稳点出现在 x = 0 与 x = 2。二阶导数为 d²y ÷ dx² = 6x − 6。当 x = 0 时其值为 −6，故 (0, 4) 为局部极大值；当 x = 2 时其值为 6，故 (2, 0) 为局部极小值。

Test | 判定	Conclusion | 结论
dy ÷ dx = 0, d²y ÷ dx² > 0	Local minimum | 局部极小值
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