Edexcel Maths: Mastering Differentiation | Edexcel 数学：微分 考点精讲

📚 Edexcel Maths: Mastering Differentiation | Edexcel 数学：微分 考点精讲

Differentiation is a fundamental pillar of Edexcel A-Level Mathematics. From basic power functions to implicit and parametric equations, mastery of this topic is essential for top exam performance. This guide covers every key technique, with step-by-step explanations and common pitfalls to avoid.

微分是 Edexcel A-Level 数学的基石。从基本的幂函数到隐函数和参数方程，掌握这一主题对于取得优异成绩至关重要。本指南涵盖每个关键技巧，提供逐步解释和常见错误避坑。

1. Power Rule & Basic Derivatives | 幂法则与基本导数

If y = xⁿ, the derivative is dy/dx = n x^n-1. This rule works for any real exponent n, including fractions and negatives.

若 y = xⁿ，则导数为 dy/dx = n x^n-1。该法则适用于任何实数指数 n，包括分数和负数。

d/dx (xⁿ) = n x^n-1

Constant multiples and sums: d/dx [c f(x)] = c f'(x) and d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

常数倍与和差法则：d/dx [c f(x)] = c f'(x)，以及 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

Example: y = 4x³ – 2x + 7 → dy/dx = 12x² – 2.

示例：y = 4x³ – 2x + 7 → dy/dx = 12x² – 2。

Do not forget that the derivative of a constant is zero.

不要忘记常数的导数为零。

2. Chain Rule | 链式法则

When a function is a composition y = f(g(x)), the chain rule states: dy/dx = f'(g(x)) · g'(x). Think ‘derivative of outer function times derivative of inner function’.

当函数为复合型 y = f(g(x))，链式法则为：dy/dx = f'(g(x)) · g'(x)。即“外函数的导数乘以内函数的导数”。

dy/dx = dy/du · du/dx

For y = (3x²+5)⁴, let u = 3x²+5. Then dy/du = 4u³, du/dx = 6x, so dy/dx = 4(3x²+5)³ · 6x = 24x(3x²+5)³.

例如 y = (3x²+5)⁴，令 u = 3x²+5。则 dy/du = 4u³，du/dx = 6x，所以 dy/dx = 4(3x²+5)³ · 6x = 24x(3x²+5)³。

Common mistake: forgetting to multiply by the derivative of the inner function.

常见错误：忘记乘以内函数的导数。

3. Product Rule | 乘积法则

If y = u(x) v(x), then dy/dx = u dv/dx + v du/dx. A handy mnemonic: ‘first times derivative of second plus second times derivative of first’.

若 y = u(x) v(x)，则 dy/dx = u dv/dx + v du/dx。容易记忆：“第一个乘第二个的导数加上第二个乘第一个的导数”。

y’ = u v’ + v u’

Example: y = x² sin x. Let u = x², v = sin x. Then u’ = 2x, v’ = cos x. So dy/dx = x² cos x + sin x · 2x.

示例：y = x² sin

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