Differentiation Rules and Applications | 微分法则与应用

📚 Differentiation Rules and Applications | 微分法则与应用

Differentiation is one of the most important topics in Edexcel A-Level Pure Mathematics. It provides a systematic way to calculate gradients, rates of change and turning points, and it underpins many modelling problems in mechanics and statistics.

微分是 Edexcel A-Level 纯数学中最重要的主题之一。它提供了一种系统的方法来计算梯度、变化率和驻点，并且是力学和统计学中许多建模问题的基础。

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1. What You Need to Know Before Differentiating | 微分前需要掌握的基础

Before you start differentiating, make sure you are confident with algebraic manipulation, laws of indices, function notation and the equation of a straight line. Many differentiation mistakes come from weak algebra rather than misunderstanding the calculus itself.

在开始学习微分之前，请确保你熟练掌握代数运算、指数律、函数记号和直线方程。许多微分错误源于代数基础薄弱，而不是对微积分本身的理解有误。

You should also be able to rewrite expressions such as 1/x as x⁻¹, √x as x¹⁄² and 1/x² as x⁻² before differentiating.

你还应该能够在求导之前将表达式改写，例如把 1/x 写成 x⁻¹，把 √x 写成 x¹⁄²，把 1/x² 写成 x⁻²。

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2. First Principles and the Gradient Function | 第一性原理与导函数

The derivative of a function f(x) is defined by the limit of the difference quotient. This is called differentiation from first principles and is sometimes examined directly.

函数 f(x) 的导数由差商的极限定义。这称为从第一性原理求导，考试中有时会直接考查。

f'(x) = lim_h→0 [f(x + h) − f(x)] / h

For example, if f(x) = x², expanding f(x + h) gives x² + 2xh + h², so the quotient becomes 2x + h, and the limit is 2x.

例如，如果 f(x) = x²，展开 f(x + h) 得到 x² + 2xh + h²，因此差商变为 2x + h，极限为 2x。

This limit gives the gradient of the tangent to the curve at a specific point, so it is often called the gradient function.

该极限给出了曲线在某一点处切线的斜率，因此它通常被称为导函数。

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3. Power, Constant and Sum Rules | 幂函数、常数与和差法则

For any real constant n, the derivative of xⁿ is nxⁿ⁻¹. This is the power rule and it is the most frequently used rule in A-Level differentiation.

对于任意实数常数 n，xⁿ 的导数是 nxⁿ⁻¹。这就是幂法则，也是 A-Level 微分中最常用的法则。

The derivative of a constant is zero, and the derivative of a sum or difference is the sum or difference of the derivatives.

常数的导数为零，和或差的导数等于各导数的和或差。

d/dx [axⁿ + bxᵐ + c] = anxⁿ⁻¹ + bmxᵐ⁻¹

Example: y = 3x⁴ − 5x² + 7x − 2 gives dy/dx = 12x³ − 10x + 7.

例子：y = 3x⁴ − 5x² + 7x − 2，得到 dy/dx = 12x³ − 10x + 7。

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4. Chain Rule | 链式法则

The chain rule is used when one function is inside another, such as y = (3x² + 5)⁴ or y = √(2x + 1). If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

链式法则用于一个函数嵌套在另一个函数中的情况，例如 y = (3x² + 5)⁴ 或 y = √(2x + 1)。如果 y = f(u) 且 u = g(x)，那么 dy/dx = dy/du × du/dx。

dy/dx = dy/du × du/dx

A practical method is to differentiate the outer function, keep the inner function unchanged, and then multiply by the derivative of the inner function.

一种实用的方法是先对外层函数求导，保持内层函数不变，然后乘以内层函数的导数。

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

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5. Product Rule | 乘积法则

When differentiating a product of two functions, y = u(x)v(x), use the product rule. Do not simply multiply the two derivatives together.

当对两个函数的乘积 y = u(x)v(x) 求导时，要使用乘积法则。不要简单地将两个导数相乘。

dy/dx = u dv/dx + v du/dx

A useful way to remember this is ‘first times derivative of second plus second times derivative of first’.

记住这个法则的一个有用方法是“第一项乘以第二项的导数加上第二项乘以第一项的导数”。

Example: y = x²eˣ. Let u = x² and v = eˣ. Then du/dx = 2x and dv/dx = eˣ, so dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2).

例子：y = x²eˣ。设 u = x²，v = eˣ。则 du/dx = 2x，dv/dx = eˣ，因此 dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2)。

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6. Quotient Rule | 商法则

For a quotient y = u(x)/v(x), the derivative is found using the quotient rule. It is especially important when the denominator is not a simple power of x.

对于商式 y = u(x)/v(x)，其导数要使用商法则来求

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