A-Level Mathematics: Differentiation Essentials | A-Level 数学：微分考点精讲

📚 A-Level Mathematics: Differentiation Essentials | A-Level 数学：微分考点精讲

Differentiation is a cornerstone of A-Level Mathematics, providing tools to analyse instantaneous rates of change, gradients, and the behaviour of functions. This revision guide covers the essential rules, techniques, and applications you need to master, from first principles to optimising real-world problems. Whether you are preparing for the Pure Mathematics paper or tackling mechanics and statistics, a firm grip on differentiation is a must.

微分是 A-Level 数学的基石，它提供了分析瞬时变化率、斜率和函数性态的工具。本复习指南涵盖了从基本原理到实际优化问题的核心法则、技巧与应用。无论你是准备纯数试卷，还是应对力学与统计，扎实掌握微分都必不可少。

1. Basic Differentiation Rules | 基本微分法则

Differentiation from first principles defines the derivative f'(x) as the limit of (f(x+h) – f(x))/h as h → 0. In practice, we use standard rules: for any real number n, the derivative of xⁿ is nxⁿ⁻¹. The derivative of a constant is zero, and constants can be factored out. Sums and differences are differentiated term by term.

从第一原理出发，导数 f'(x) 定义为当 h → 0 时 (f(x+h) – f(x))/h 的极限。实际计算中我们使用标准法则：对任意实数 n，xⁿ 的导数为 nxⁿ⁻¹。常数的导数为零，常数因子可以提出，和与差逐项微分。

Function f(x)	Derivative f'(x)
xⁿ	n xⁿ⁻¹
k (constant)	0
k · g(x)	k · g'(x)
g(x) ± h(x)	g'(x) ± h'(x)

上面的表格总结了最基本的微分法则。牢记这些公式是解决一切微分问题的起点，后续所有技巧都建立在这个基础之上。

2. The Chain Rule | 链式法则

The chain rule handles composite functions. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. In function notation, (f(g(x)))’ = f'(g(x)) · g'(x). It is often written as: differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside.

链式法则处理复合函数：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。用函数记号即 (f(g(x)))’ = f'(g(x)) · g'(x)。通常的口诀是“外导乘内导”：先对外层函数求导并保留内层函数，再乘上内层函数的导数。

Example: y = (3x² – 5)⁴. Let u = 3x² – 5, y = u⁴. Then dy/du = 4u³, du/dx = 6x. So dy/dx = 4(3x² – 5)³ × 6x = 24x(3x² – 5)³. The chain rule is indispensable for exponentials, logarithms, and trigonometric functions of linear or more complicated arguments.

例：y = (3x² – 5)⁴，设 u = 3x² – 5，y = u⁴，则 dy/du = 4u³，du/dx = 6x，所以 dy/dx = 4(3x² – 5)³ × 6x = 24x(3x² – 5)³。链式法则在指数、对数以及线性或更复杂内层的三角函数中必不可少。

3. Product and Quotient Rules | 乘积法则与商法则

When two differentiable functions multiply or divide, we need special rules. The product rule: if y = u(x)v(x), then y’ = u’v + uv’. The quotient rule: if y = u(x)/v(x), then y’ = (u’v – uv’) / v². Careful ordering matters in the quotient rule, and the square of the denominator is always positive, so the sign of the derivative depends on the numerator.

当两个可微函数相乘或相除时，需要使用专门的法则。乘积法则：若 y = u(x)v(x)，则 y’ = u’v + uv’。商法则：若 y = u(x)/v(x)，则 y’ = (u’v – uv’) / v²。商法则中分子顺序十分重要，分母的平方恒正，因此导数的符号完全取决于分子。

Product Rule: d/dx [u v] = u’ v + u v’

乘积法则：d/dx [u v] = u’ v + u v’

Quotient Rule: d/dx [u/v] = (u’ v – u v’) / v²

商法则：d/dx [u/v] = (u’ v – u v’) / v²

For example, y = x² eˣ: set u = x², v = eˣ. Then u’ = 2x, v’ = eˣ, so y’ = 2x eˣ + x² eˣ = x eˣ (2 + x). For y = sin x / x, u = sin x, v = x, u’ = cos x, v’ = 1, so y’ = (x cos x – sin x) / x².

例如 y = x² eˣ：令 u = x², v = eˣ，则 u’ = 2x, v’ = eˣ，y’ = 2x eˣ + x² eˣ = x eˣ (2 + x)。对 y = sin x / x，u = sin x, v = x, u’ = cos x, v’ = 1，y’ = (x cos x – sin x) / x²。

4. Differentiation of Exponential and Logarithmic Functions | 指数与对数函数的微分

The natural exponential function eˣ is its own derivative: d/dx (eˣ) = eˣ. More generally, d/dx (aˣ) = aˣ ln a for a > 0. The natural logarithm ln x differentiates to 1/x (x > 0). For logarithms with other bases, d/dx (logₐ x) = 1/(x ln a). These are often combined with the chain rule when the exponent or argument is a function of x.

自然指数函数 eˣ 的导数就是它自身：d/dx (eˣ) = eˣ。更一般地，对 a > 0 有 d/dx (aˣ) = aˣ ln a。自然对数 ln x 的导数为 1/x（x > 0）。其他底数的对数，d/dx (logₐ x) = 1/(x ln a)。当指数或对数内部是 x 的函数时，常与链式法则结合使用。

d/dx (eᶻ) = eᶻ · dz/dx, d/dx (ln z) = (1/z) dz/dx

d/dx (eᶻ) = eᶻ · dz/dx， d/dx (ln z) = (1/z) dz/dx

Example: y = eˣ². Using chain rule, dz/dx = 2x, so dy/dx = eˣ² · 2x = 2x eˣ². For y = ln(5x³), derivative is (1/(5x³)) · 15x² = 3/x. These rules are fundamental for many integration and modelling problems.

例：y = eˣ²，利用链式法则，dz/dx = 2x，故 dy/dx = eˣ² · 2x = 2x eˣ²。对 y = ln(5x³)，导数为 (1/(5x³)) · 15x² = 3/x。这些法则在众多积分与建模问题中至关重要。

5. Trigonometric Differentiation | 三角函数的微分

Differentiating trigonometric functions requires remembering the standard derivatives. All circular functions can be differentiated using these base results together with the chain, product, and quotient rules.

三角函数的微分需要牢记标准导数。所有圆函数都可以通过这些基本结果并配合链式、乘积和商法则进行求导。

d/dx (sin x) = cos x
d/dx (cos x) = –sin x
d/dx (tan x) = sec² x
d/dx (sec x) = sec x tan x
d/dx (csc x) = –csc x cot x
d/dx (cot x) = –csc² x

上面列出了六个三角函数的导数公式。请注意符号和倒数函数的关系，例如 sec x 的导数为 sec x tan x，csc x 的导数为 –csc x cot x，这些在复杂应用题中经常出现。

For instance, differentiate y = sin(2x). By the chain rule, dy/dx = 2 cos(2x). For y = tan² x, using the chain rule with u = tan x, dy/dx = 2 tan x · sec² x. These patterns appear frequently in A-Level questions involving stationary points or integration.

例如，对 y = sin(2x) 求导，利用链式法则得 dy/dx = 2 cos(2x)。对 y = tan² x，令 u = tan x 并用链式法则，dy/dx = 2 tan x · sec² x。这些模式在涉及驻点或积分的 A-Level 考题中频繁出现。

6. Implicit Differentiation | 隐函数微分

When a relationship between x and y is given implicitly, such as x² + y² = 25, we cannot easily write y = f(x). Implicit differentiation treats y as a function of x and applies the chain rule: every time we differentiate a term in y, we multiply by dy/dx. Then we rearrange to isolate dy/dx.

当 x 和 y 的关系以隐式给出，例如 x² + y² = 25，我们无法轻易写成 y = f(x) 的形式。隐函数微分将 y 视为 x 的函数并运用链式法则：每对 y 的项求导，就要乘以 dy/dx。然后通过移项解出 dy/dx。

Given x² + y² = 25, differentiating: 2x + 2y (dy/dx) = 0 → dy/dx = –x/y

对于 x² + y² = 25，微分得 2x + 2y (dy/dx) = 0 → dy/dx = –x/y

This technique is vital for finding tangents and normals to curves like circles, ellipses, or more complicated relations. Always remember to apply the product rule when terms involve products of x and y, such as xy, and to carefully handle any powers of y via the chain rule.

这种方法在求圆、椭圆或其他复杂关系的切线与法线时至关重要。务必注意，遇到 x 与 y 的乘积项（如 xy）时要使用乘积法则，对 y 的幂次则通过链式法则处理。

7. Parametric Differentiation | 参数方程的微分

Parametric equations define x and y separately in terms of a third variable t: x = f(t), y = g(t). The gradient dy/dx is given by (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This arises directly from the chain rule: dy/dt = dy/dx · dx/dt.

参数方程通过第三个变量 t 分别定义 x 和 y：x = f(t)，y = g(t)。斜率 dy/dx 由 (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。这一结果直接来自链式法则：dy/dt = dy/dx · dx/dt。

dy/dx = (dy/dt) / (dx/dt)

To find the second derivative d²y/dx², we differentiate dy/dx with respect to x, using the chain rule again: d²y/dx² = d/dt (dy/dx) ÷ dx/dt. This is a common pitfall – do not simply divide d²y/dt² by d²x/dt².

为了求二阶导数 d²y/dx²，需对 dy/dx 关于 x 求导，再次使用链式法则：d²y/dx² = d/dt (dy/dx) ÷ dx/dt。这是一个常见陷阱——切勿将 d²y/dt² 直接除以 d²x/dt²。

d²y/dx² = [d/dt (dy/dx)] / (dx/dt)

Example: x = t², y = 2t. Then dx/dt = 2t, dy/dt = 2, so dy/dx = 2/(2t) = 1/t. For the second derivative, d/dt (1/t) = –1/t², so d²y/dx² = (–1/t²) / (2t) = –1/(2t³).

例：x = t², y = 2t，则 dx/dt = 2t, dy/dt = 2，故 dy/dx = 2/(2t) = 1/t。二阶导数：d/dt (1/t) = –1/t²，d²y/dx² = (–1/t²) / (2t) = –1/(2t³)。

8. Higher-Order Derivatives | 高阶导数

The first derivative f'(x) represents the rate of change of a function. The second derivative f”(x) or d²y/dx² gives the rate of change of the gradient, which tells us about the function’s concavity and helps classify stationary points. Third, fourth, and higher derivatives are occasionally used, but the second derivative is the most common beyond the first.

一阶导数 f'(x) 表示函数的变化率。二阶导数 f”(x) 或 d²y/dx² 则是梯度变化率，它揭示了函数的凹凸性，并用于分类驻点。三阶、四阶及更高阶导数偶尔会用到，但二阶导数在一阶之外最常用。

To find higher derivatives, simply differentiate repeatedly. For y = x⁴, y’ = 4x³, y” = 12x², y”’ = 24x, y⁽⁴⁾ = 24, and y⁽⁵⁾ = 0. In kinematics, if s is displacement, then v = ds/dt (velocity) and a = d²s/dt² (acceleration). Remember that notation like f⁽ⁿ⁾(x) indicates the nth derivative.

求高阶导数只需反复微分即可。对于 y = x⁴，y’ = 4x³，y” = 12x²，y”’ = 24x，y⁽⁴⁾ = 24，y⁽⁵⁾ = 0。在运动学中，若 s 表示位移，则 v = ds/dt（速度）而 a = d²s/dt²（加速度）。记住记号 f⁽ⁿ⁾(x) 表示 n 阶导数。

9. Tangents, Normals, and Stationary Points | 切线、法线与驻点

At a point (x₀, y₀) on a curve y = f(x), the gradient of the tangent is m = f'(x₀). Its equation is y – y₀ = m(x – x₀). The normal is perpendicular to the tangent, so its gradient is –1/m, and its equation is y – y₀ = (–1/m)(x – x₀). Stationary points occur where f'(x) = 0, indicating a possible maximum, minimum, or point of inflection.

在曲线 y = f(x) 上一点 (x₀, y₀) 处，切线斜率为 m = f'(x₀)，其方程为 y – y₀ = m(x – x₀)。法线与切线垂直，故斜率为 –1/m，方程为 y – y₀ = (–1/m)(x – x₀)。驻点出现在 f'(x) = 0 处，可能是极大值点、极小值点或拐点。

To classify a stationary point at x = a, use the second derivative test: if f”(a) > 0, it’s a local minimum; if f”(a) < 0, it's a local maximum; if f''(a) = 0, the test is inconclusive and we examine the sign change of f'(x) around x = a. Understanding the shape of the graph through concavity is key.

要分类 x = a 处的驻点，使用二阶导数判别法：若 f”(a) > 0，则为局部极小值；若 f”(a) < 0，则为局部极大值；若 f''(a) = 0，该判别法失效，需检查 x = a 两侧 f'(x) 的符号变化。通过凹凸性理解图像形状是关键。

Example: f(x) = x³ – 3x, f'(x) = 3x² – 3 = 0 gives x = ±1. f”(x) = 6x, so at x = 1, f”(1) = 6 > 0 → local minimum; at x = –1, f”(–1) = –6 < 0 → local maximum. Always find the corresponding y-coordinates for full coordinates of stationary points.

例：f(x) = x³ – 3x，f'(x) = 3x² –

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A-Level Mathematics: Differentiation Essentials | A-Level 数学：微分考点精讲