Advanced Differentiation Techniques for Edexcel A-Level Pure Mathematics | Edexcel A-Level 纯数学进阶微分技巧

📚 Advanced Differentiation Techniques for Edexcel A-Level Pure Mathematics | Edexcel A-Level 纯数学进阶微分技巧

Differentiation is one of the most powerful tools in A-Level mathematics. This article covers advanced techniques required for Edexcel Pure Mathematics, including the chain, product and quotient rules, implicit and parametric differentiation, and applications such as stationary points and connected rates of change.

微分是 A-Level 数学中最强大的工具之一。本文涵盖 Edexcel 纯数学所需的高级技巧，包括链式法则、乘积法则和商法则、隐函数与参数方程求导，以及驻点和关联变化率等应用。

1. Review of Basic Differentiation Rules | 基本微分规则回顾

Before moving to advanced techniques, ensure you can differentiate powers, constant multiples, sums and differences without hesitation. The general power rule for any real n is given below.

在学习高级技巧之前，请确保你能毫不犹豫地对幂函数、常数倍、和与差进行求导。对任意实数 n 的一般幂法则如下。

d/dx (xⁿ) = n xⁿ⁻¹

You should also remember that the derivative of a constant is zero, and the derivative of a sum is the sum of the derivatives: d/dx [f(x) + g(x)] = f'(x) + g'(x).

你还应记住常数的导数为零，且和的导数等于导数之和：d/dx [f(x) + g(x)] = f'(x) + g'(x)。

These rules allow you to differentiate polynomials term by term, such as y = 4x³ – 2x² + 7x – 5, giving dy/dx = 12x² – 4x + 7.

这些法则使你能够逐项求导多项式，例如 y = 4x³ – 2x² + 7x – 5，得到 dy/dx = 12x² – 4x + 7。

2. The Chain Rule | 链式法则

The chain rule is used when one function is nested inside another. If y = f(u) and u = g(x), then dy/dx = dy/du × du

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