📚 Differentiation Rules for Edexcel A-Level Pure Maths | Edexcel A-Level 纯数学微分法则精讲
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It underpins applications ranging from tangents and normals to kinematics, optimisation and curve sketching. This revision guide brings together the core rules, examples and exam tips you need to master differentiation with confidence.
微分是 Edexcel A-Level 纯数学中考查频率最高的内容之一。从切线与法线到运动学、优化问题和曲线草图，微分都是核心工具。本复习指南汇总核心法则、例题与应试技巧，帮助你自信掌握微分。
1. Gradients and Differentiation from First Principles | 梯度与第一性原理求导
Differentiation measures the instantaneous rate of change of a function. For a curve y = f(x), the gradient of the tangent at x = a is found using the limit f'(a) = limₕ→₀ [f(a+h) − f(a)] / h. This definition is called differentiation from first principles and is a required Edexcel skill.
微分度量函数的瞬时变化率。对曲线 y = f(x)，x = a 处切线的斜率可由极限 f'(a) = limₕ→₀ [f(a+h) − f(a)] / h 求得。这个定义称为第一性原理求导，是 Edexcel 考纲要求学生掌握的技能。
In exam questions, first principles usually appear with simple polynomials such as x² or x³. Always show the expansion, cancellation of h and the limiting step clearly to secure full marks.
考试中第一性原理常以 x² 或 x³ 等简单多项式出现。务必清晰展示展开、约去 h 以及取极限的过程，才能获得满分。
2. The Power Rule and Constant Multiple Rule | 幂函数法则与常数倍法则
For any real constant n, the derivative of xⁿ is n xⁿ⁻¹. Constants factor out, and the derivative of a sum is the sum of the derivatives.
对任意实常数 n，xⁿ 的导数为 n xⁿ⁻¹。常数可以提出，和的导数等于各项导数之和。
d/dx (xⁿ) = n xⁿ⁻¹ and d/dx [k f(x)] = k f'(x)
Negative and fractional powers work in exactly the same way. For example, d/dx (1/x²) = d/dx (x⁻²) = −2x⁻³, and d/dx (√x) = d/dx (x^½) = ½ x^−½.
负指数和分数指数同样适用。例如 d/dx (1/x²) = d/dx (x⁻²) = −2x⁻³，且 d/dx (√x) = d/dx (x^½) = ½ x^−½。
3. Chain Rule for Composite Functions | 复合函数链式法则
If y = g(f(x)), then dy/dx = g'(f(x)) × f'(x). In Leibniz notation, if y = g(u) and u = f(x), then dy/dx = dy/du × du/dx. This rule is essential for functions such as (3x² + 5)⁷ or sin(2x).
若 y = g(f(x))，则 dy/dx = g'(f(x)) × f'(x)。用莱布尼茨记号，若 y = g(u) 且 u = f(x)，则 dy/dx = dy/du × du/dx。处理 (3x² + 5)⁷ 或 sin(2x) 这类函数时链式法则至关重要。
dy/dx = dy/du × du/dx
Example: if y = (3x² + 5)⁷, let u = 3x² + 5. Then dy/du = 7u⁶ and du/dx = 6x, so dy/dx = 7(3x² + 5)⁶ × 6x = 42x(3x² + 5)⁶.
例题：若 y = (3x² + 5)⁷，令 u = 3x² + 5，则 dy/du = 7u⁶，du/dx = 6x，因此 dy/dx = 7(3x² + 5)⁶ × 6x = 42x(3x² + 5)⁶。
4. Product Rule | 乘积法则
If y = u v, where u and v are functions of x, then dy/dx = u dv/dx + v du/dx. In words: first times derivative of second, plus second times derivative of first.
若 y = u v，其中 u 和 v 都是 x 的函数，则 dy/dx = u dv/dx + v du/dx。简单说：第一项乘以第二项的导数，加上第二项乘以第一项的导数。
d/dx (u v) = u v’ + v u’
Example: y = x² eˣ. Let u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, so dy/dx = x² eˣ + 2x eˣ = x eˣ (x + 2).
例题：y = x² eˣ。令 u = x²，v = eˣ，则 u’ = 2x，v’ = eˣ，因此 dy/dx = x² eˣ + 2x eˣ = x eˣ (x + 2)。
5. Quotient Rule | 商法则
If y = u / v, then dy/dx = (v du/dx − u dv/dx) / v². The numerator is ‘v du minus u dv’, not the other way round; reversing the order is a frequent sign error.
若 y = u / v，则 dy/dx = (v du/dx − u dv/dx) / v²。分子是“v du 减 u dv”，不要颠倒顺序；顺序反了是常见的符号错误。
d/dx (u/v) = (v u’ − u v’) / v²
Example: y = x / (x² + 1). Let u = x and v = x² + 1. Then u’ = 1 and v’ = 2x, so dy/dx = [(x² + 1)(1) − x(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)².
例题：y = x / (x² + 1)。令 u = x，v = x² + 1，则 u’ = 1，v’ = 2x，因此 dy/dx = [(x² + 1)(1) − x(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)²。
6. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The exponential function eˣ is its own derivative. More generally, d/dx (aˣ) = aˣ ln a. For natural logarithms, d/dx (ln x) = 1/x, and by the chain rule d/dx (ln|f(x)|) = f'(x) / f(x).
指数函数 eˣ 的导数就是它本身。更一般地，d/dx (aˣ) = aˣ ln a。对自然对数，d/dx (ln x) = 1/x，由链式法则可得 d/dx (ln|f(x)|) = f'(x) / f(x)。
d/dx (eˣ) = eˣ and d/dx (ln x) = 1/x
Example: if y = ln(x² + 1), then dy/dx = 2x / (x² + 1). If y = e^(3x), then dy/dx = 3 e^(3x).
例题：若 y = ln(x² + 1)，则 dy/dx = 2x / (x² + 1)。若 y = e^(3x)，则 dy/dx = 3 e^(3x)。
7. Differentiating Trigonometric Functions | 三角函数的微分
Standard trigonometric derivatives are used frequently in Edexcel questions. Learn them actively rather than deriving them in the exam.
标准三角函数的导数在 Edexcel 考题中频繁出现。建议主动熟记，而不是在考场上临时推导。
| d/dx (sin x) = cos x | sin x 的导数为 cos x |
| d/dx (cos x) = −sin x | cos x 的导数为 −sin x |
| d/dx (tan x) = sec² x | tan x 的导数为 sec² x |
| d/dx (sec x) = sec x tan x | sec x 的导数为 sec x tan x |
| d/dx (cosec x) = −cosec x cot x | cosec x 的导数为 −cosec x cot x |
| d/dx (cot x) = −cosec² x | cot x 的导数为 −cosec² x |
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