Edexcel A Level Pure Maths: Differentiation Techniques and Applications | 爱德思A Level纯数学：微分技巧与应用

📚 Edexcel A Level Pure Maths: Differentiation Techniques and Applications | 爱德思A Level纯数学：微分技巧与应用

This revision guide covers the differentiation toolkit required for Edexcel A Level Mathematics, including standard derivatives, the chain, product, and quotient rules, exponential, logarithmic, and trigonometric derivatives, parametric differentiation, stationary points, tangents and normals, and connected rates of change. Work through each section with pencil and paper, and check every formula by substituting a simple function.

本复习指南涵盖爱德思A Level数学微分部分所需的全部核心工具，包括标准导数、链式法则、乘积法则、商法则、指数函数、对数函数和三角函数的导数、参数方程求导、驻点、切线与法线以及相关变化率。建议准备好纸笔，逐节练习，并代入简单函数检验每个公式。

1. Gradient Function and Standard Derivatives | 梯度函数与标准导数

The derivative f'(x) or dy/dx measures the instantaneous rate of change of y with respect to x, and gives the gradient of the tangent to the curve y = f(x). For any power function y = xⁿ, where n is a real constant, the derivative is dy/dx = n xⁿ⁻¹. This rule applies to positive integers, negative powers, and fractional powers such as √x = x^(1/2).

导数 f'(x) 或 dy/dx 表示 y 关于 x 的瞬时变化率，也等于曲线 y = f(x) 切线的斜率。对任意幂函数 y = xⁿ（n 为实数常数），其导数为 dy/dx = n xⁿ⁻¹。这一法则既适用于正整数次幂，也适用于负指数和分数指数，例如 √x = x^(1/2)。

d/dx(xⁿ) = n xⁿ⁻¹

For example, d/dx(x⁵) = 5x⁴, d/dx(x⁻²) = -2x⁻³, and d/dx(√x) = 1/(2√x). Constant multiples and sums differentiate term by term: d/dx[3x² + 4x – 7] = 6x + 4.

例如，d/dx(x⁵) = 5x⁴，d/dx(x⁻²) = -2x⁻³，d/dx(√x) = 1/(2√x)。常数倍与和可以逐项求导：d/dx[3x² + 4x – 7] = 6x + 4。

2. Chain Rule | 链式法则

The chain rule is used when a function is a composition, y = f(u) where u = g(x). Then dy/dx = (dy/du) × (du/dx). In function notation, if y = [g(x)]ⁿ, then dy/dx = n[g(x)]ⁿ⁻¹ × g'(x). For example, y = (2x+3)⁴ gives dy/dx = 4(2x+3)³ × 2 = 8(2x+3)³.

链式法则用于复合函数：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。用函数记号表示，如果 y = [g(x)]ⁿ，则 dy/dx = n[g(x)]ⁿ⁻¹ × g'(x)。例如，y = (2x+3)⁴ 的导数为 dy/dx = 4(2x+3)³ × 2 = 8(2x+3)³。

dy/dx = (dy/du) × (du/dx)

Identify the inside function u first, differentiate u with respect to x, differentiate y with respect to u, then multiply the two results. This method also works for functions such as y = √(x² + 1) by rewriting the square root as a power of one half.

先识别内层函数 u，对 u 关于 x 求导，再对 y 关于 u 求导，最后将两个结果相乘。该方法也适用于 y = √(x² + 1) 这类函数，只需先把平方根写成二分之一次幂。

3. Product Rule | 乘积法则

For y = u(x)v(x), the product rule states dy/dx = u dv/dx + v du/dx. It is useful when two non-constant functions are multiplied. For example, y = x² sin x has derivative dy/dx = 2x sin x + x² cos x. Always write u and v clearly before differentiating.

对于 y = u(x)v(x)，乘积法则是 dy/dx = u dv/dx + v du/dx。当两个非常数函数相乘时使用该法则。例如，y = x² sin x 的导数为 dy/dx = 2x sin x + x² cos x。在求导前务必先写出 u 和 v。

d/dx(uv) = u dv/dx + v du/dx

Do not multiply the two functions together before differentiating unless the expansion is simple. The product rule is often combined with the chain rule when one factor is a composite function, such as y = x e²ˣ.

除非展开后非常简单，否则不要先相乘再求导。当一个因子是复合函数时，乘积法则通常与链式法则结合使用，例如 y = x e²ˣ。

4. Quotient Rule | 商法则

For y = u(x)/v(x), the quotient rule states dy/dx = [v du/dx – u dv/dx] / v². The order is critical: v du comes first. For example, y = x/(x+1) gives dy/dx = [(x+1)(1) – x(1)]/(x+1)² = 1/(x+1)². Simplify the numerator before cancelling.

对于 y = u(x)/v(x)，商法则是 dy/dx = [v du/dx – u dv/dx] / v²。顺序非常关键：先 v 乘 u 的导数。例如，y = x/(x+1) 的导数为 dy/dx = [(x+1)(1) – x(1)]/(x+1)² = 1/(x+1)²。在约分前先化简分子。

d/dx(u/v) = (v du/dx – u dv/dx) / v²

A common exam error is writing u dv/dx first in the numerator, which gives the wrong sign. Also remember to square the denominator fully, including any constants or brackets. The quotient rule can be avoided by

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