The Quotient Rule | 商法则

📚 The Quotient Rule | 商法则

In A-Level Edexcel Mathematics, differentiation questions often present a function as a fraction. The quotient rule gives a direct way to differentiate y = u/v, where u and v are both functions of x. This lesson covers the formula, its derivation, worked examples, common mistakes, and exam tips.

在 A-Level Edexcel 数学中，微分题常以分式给出函数。商法则提供了直接对 y = u/v 求导的方法，其中 u 和 v 都是 x 的函数。本课涵盖公式、推导、例题、常见错误和考试技巧。

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1. What the Quotient Rule Is | 什么是商法则

The quotient rule is a differentiation technique used when a function is given as one expression divided by another. If y = u/v, where both u and v are functions of x, the rule gives the derivative of y without expanding or simplifying first.

商法则是一种微分技巧，用于处理一个表达式除以另一个表达式的函数。如果 y = u/v，其中 u 和 v 都是 x 的函数，该法则可以直接求出 y 的导数，而不需要先展开或化简。

In A-Level Edexcel questions, the quotient appears in forms such as rational functions, trigonometric fractions, and combinations of eˣ and ln x. You must be able to identify u and v quickly.

在 A-Level Edexcel 考题中，商的形式常见于有理函数、三角分式以及 eˣ 与 ln x 的组合。你必须能够快速识别 u 和 v。

If y = u/v, then dy/dx = (v du/dx − u dv/dx) / v²

The numerator is ‘bottom times derivative of top minus top times derivative of bottom’, and the denominator is the bottom squared.

分子是 ‘下方乘以上方的导数 减去 上方乘以下方的导数’，分母是下方的平方。

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2. The Formula and Notation | 公式与记号

Edexcel questions may use function notation or Leibniz notation. Both forms are equivalent: if y = f(x)/g(x), then y’ = (g f’ − f g’) / g².

Edexcel 题目可能使用函数记号或莱布尼茨记号。两种形式等价：如果 y = f(x)/g(x)，则 y’ = (g f’ − f g’) / g²。

You can also write the rule as dy/dx = (v · du/dx − u · dv/dx) / v². The multiplication dots help keep the structure clear when substituting values.

你也可以写成 dy/dx = (v · du/dx − u · dv/dx) / v²。代入数值时，乘号有助于保持结构清晰。

Here, v is the denominator of the original fraction and u is the numerator. Do not confuse their order: the term with v du/dx comes first.

这里，v 是原分式的分母，u 是分子。不要混淆顺序：含有 v du/dx 的项在前。

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3. When to Use It | 何时使用

Use the quotient rule when the function is clearly written as a fraction with a non-constant denominator. For example, y = (x² + 3)/(x − 1) is a natural candidate.

当函数明显写成分式且分母不是常数时，使用商法则。例如 y = (x² + 3)/(x − 1) 就适合使用。

Sometimes simplifying first is easier. For instance, y = (x² + 2x)/x can be rewritten as y = x + 2, so its derivative is simply 1. Using the quotient rule here would still work, but it is longer.

有时先化简更容易。例如 y = (x² + 2x)/x 可以写成 y = x + 2，因此导数就是 1。这里用商法则也能算，但过程更长。

In an exam, check the form of the function before choosing a method. If the denominator is a constant, use the constant multiple rule instead.

考试时，在选择方法之前先检查函数形式。如果分母是常数，使用常数倍法则即可。

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4. Deriving from the Product Rule | 从乘积法则推导

The quotient rule can be derived by writing y = u/v as y = u · v⁻¹ and applying the product rule and chain rule.

商法则可以通过将 y = u/v 写成 y = u · v⁻¹，再应用乘积法则和链式法则来推导。

dy/dx = du/dx · v⁻¹ + u · (−v⁻² · dv/dx)

This simplifies to dy/dx = (v du/dx − u dv/dx) / v², which is the quotient rule.

这可以化简为 dy/dx = (v du/dx − u dv/dx) / v²，正是商法则。

This derivation helps explain why the sign is negative in the rule: the negative exponent in v⁻¹ produces a minus sign when differentiating v.

这个推导有助于解释为什么法则中有负号：v⁻¹ 中的负指数在对 v 求导时产生了负号。

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5. Worked Example: Polynomial Quotient | 例题：多项式商

Differentiate y = (x² + 1)/(3x − 2).

求 y = (x² + 1)/(3x − 2) 的导数。

Let u = x² + 1, so du/dx = 2x. Let v = 3x − 2, so dv/dx = 3.

设 u = x² + 1，则 du/dx = 2x。设 v = 3x − 2，则 dv/dx = 3。

dy/dx = ((3x − 2)(2x) − (x² + 1)(3)) / (3x − 2)²

dy/dx = (6x² − 4x − 3x² − 3) / (3x − 2)² = (3x² − 4x − 3) / (3x − 2)²

Always expand the numerator carefully, but leave the denominator in its squared form unless the question asks for further simplification.

务必仔细展开分子，但除非题目要求进一步化简，否则分母保持

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