Chapter 9: Differentiation | 第9章：微分

📚 Chapter 9: Differentiation | 第9章：微分

Differentiation is one of the two central ideas in calculus and appears throughout the Edexcel A Level Mathematics specification. It measures the instantaneous rate of change of a function and provides the tools needed for tangents, normals, stationary points, optimisation and modelling.

微分是微积分的两大核心思想之一，贯穿 Edexcel A-Level 数学大纲。它度量函数在某一时刻的变化率，为切线、法线、驻点、最优化和数学建模提供必要工具。

1. Differentiating from First Principles | 从第一原理求导

The derivative is defined by a limit: f'(x) = lim (h→0) [f(x+h) − f(x)] / h. You may be asked to use this definition to differentiate simple functions such as x² or 1/x.

导数由一个极限定义：f'(x) = lim (h→0) [f(x+h) − f(x)] / h。考试中可能要求你利用该定义求简单函数（如 x² 或 1/x）的导数。

For f(x) = x², expand (x+h)² = x² + 2xh + h², subtract x², divide by h and let h → 0 to obtain f'(x) = 2x. This shows the limit process behind differentiation.

对于 f(x) = x²，展开 (x+h)² = x² + 2xh + h²，减去 x² 后除以 h，再令 h → 0，得到 f'(x) = 2x。这展示了求导背后的极限过程。

f'(x) = lim (h→0) [f(x+h) − f(x)] / h

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2. Standard Derivatives | 基本导数公式

Memorising the standard derivatives is essential. For any rational power n, d/dx (xⁿ) = n xⁿ⁻¹. The derivative of eˣ is eˣ itself, while ln x gives 1/x.

熟记基本导数公式至关重要。对任意有理指数 n，d/dx (xⁿ) = n xⁿ⁻¹。eˣ 的导数仍是 eˣ，而 ln x 的导数为 1/x。

f(x)	f'(x)
xⁿ	n xⁿ⁻¹
sin x	cos x
cos x	−sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x
aˣ	aˣ ln a

These standard results are the building blocks for all harder differentiation questions. Edexcel papers often test them inside chain, product or quotient rule problems.

这些基本结果是所有较难求导问题的基础。Edexcel 试卷常将它们隐藏在链式法则、乘积法则或商法则的问题中考查。

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3. Chain Rule | 链式法则

The chain rule is used for composite functions: if y = g(f(x)), then dy/dx = g'(f(x)) × f'(x). In Leibniz notation, dy/dx = dy/du × du/dx.

链式法则用于复合函数：若 y = g(f(x))，则 dy/dx = g'(f(x)) × f'(x)。用莱布尼茨记号可写为 dy/dx = dy/du × du/dx。

Example: y = (3x² + 5)⁴. Let u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 24x(3x²+5)³.

例子：y = (3x² + 5)⁴。令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x，所以 dy/dx = 24x(3x²+5)³。

dy/dx = dy/du × du/dx

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4. Product Rule | 乘积法则

For y = u(x)v(x), the product rule is dy/dx = u’v + uv’. Always identify u, v, u’ and v’ before substituting.

对于 y = u(x)v(x)，乘积法则为 dy/dx = u’v + uv’。代入前一定要先写出 u、v、u’ 和 v’。

Example: y = x² sin x. Here u = x², v = sin x, so dy/dx = 2x sin x + x² cos x.

例子：y = x² sin x。这里 u = x²，v = sin x，因此 dy/dx = 2x sin x + x² cos x。

dy/dx = u’v + uv’

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5. Quotient Rule | 商法则

For y = u/v, the quotient rule is dy/dx = (vu’ − uv’) / v². It is useful when the denominator is not a simple power.

对于 y = u/v，商法则为 dy/dx = (vu’ − uv’) / v²。当分母不是简单幂函数时，该法则非常有用。

Example: y = (x² + 1)/(x − 2). Here u = x²+1, v = x−2, so dy/dx = [(x−2)(2x) − (x²+1)(1)] / (x−2)² = (x² − 4x − 1)/(x−2)².

例子：y = (x² + 1)/(x − 2)。这里 u = x²+1，v = x−2，因此 dy/dx = [(x−2)(2x) − (x²+1)(1)] / (x−2)² = (x² − 4x − 1)/(x−2)²。

dy/dx = (vu’ − uv’) / v²

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6. Implicit Differentiation | 隐函数求导

When an equation involves x and y and y is not explicitly the subject, differentiate both sides with respect to x. Whenever you differentiate a function of y, multiply by dy/dx.

当方程含有 x 和 y 且 y 不是显式表达时，对两边关于 x 求导。每当对 y 的函数求导，都要乘以 dy/dx。

Example: x² + y² = 25. Differentiating gives 2x + 2y dy/dx = 0, hence dy/dx = −x/y.

例子：x² + y² = 25。求导得 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。

Implicit differentiation is especially common with equations of circles and other conic sections in Edexcel past papers.

在 Edexcel 历年真题中，隐函数求导在圆和其他圆锥曲线方程中尤为常见。

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7. Parametric Differentiation | 参数方程求导

If x = f(t) and y = g(t), then dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This is used for curves such as circles, ellipses and projectile paths.

若 x = f(t) 且 y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。这用于圆、椭圆和抛体轨迹等曲线。

Example: x = t², y = 2t + 1. Then dx/dt = 2t and dy/dt = 2, so dy/dx =

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