The Chain Rule | 链式法则

📚 The Chain Rule | 链式法则

The chain rule is one of the most important differentiation techniques in A-Level mathematics. It allows us to differentiate composite functions, which are functions inside other functions. This rule appears in almost every differentiation problem and is essential for topics such as connected rates of change and implicit differentiation.

链式法则是 A-Level 数学中最重要的一项求导技巧。它让我们能够对复合函数进行求导，即函数嵌套函数的形式。这条法则几乎出现在所有求导题目中，也是学习相关变化率和隐函数求导的基础。

1. Introduction to the Chain Rule | 链式法则简介

A composite function is a function that contains another function. For example, y = sin(2x) is a composite function because the sine function is applied to the function 2x. Similarly, y = (3x² + 1)⁴ is composite because the power function wraps around the quadratic expression.

复合函数是指一个函数包含另一个函数。例如，y = sin(2x) 就是一个复合函数，因为正弦函数作用于函数 2x 上。同样地，y = (3x² + 1)⁴ 也是复合函数，因为幂函数包裹着二次表达式。

The chain rule states that if y = f(u) and u = g(x), then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.

链式法则指出：如果 y = f(u) 且 u = g(x)，那么 y 对 x 的导数等于 y 对 u 的导数与 u 对 x 的导数之积。

2. The Formula | 公式形式

There are two common ways to write the chain rule. The first uses the Leibniz notation:

链式法则有两种常见的书写方式。第一种使用莱布尼茨记号：

dy/dx = (dy/du) × (du/dx)

The second form is written directly in terms of the functions f and g:

第二种形式直接用函数 f 和 g 来表达：

d/dx [f(g(x))] = f ‘(g(x)) × g ‘(x)

The first form is often easier to remember and apply: differentiate the outer function, multiply by the derivative of the inner function.

第一种形式更容易记忆和应用：先对外层函数求导，再乘以内层函数的导数。

For example, consider y = (2x + 3)⁷. Here u = 2x + 3, so y = u⁷. Then dy/du = 7u⁶ and du/dx = 2. Therefore:

例如，考虑 y = (2x + 3)⁷。这里 u = 2x + 3，所以 y = u⁷。于是 dy/du = 7u⁶，du/dx = 2。因此：

dy/dx = 7(2x + 3)⁶ × 2 = 14(2x + 3)⁶

3. Differentiating Composite Functions | 复合函数求导

The chain rule is also known as the “function of a function” rule. When you see a function inside another, the chain rule tells you to tackle the problem in steps: differentiate the outer layer first, leaving the inner function untouched, then multiply by the derivative of the inner function.

链式法则也被称为”函数的函数”法则。当你看到函数嵌套时，链式法则指导你分步求解：先对外层求导，保持内层函数不变，然后乘以内层函数的导数。

Consider y = (x² + 5x)³. The outer function is cubing, and the inner function is x² + 5x. Differentiating the outer function gives 3(x² + 5x)². Differentiating the inner function gives 2x + 5. Hence:

考虑 y = (x² + 5x)³。外层函数是三次方，内层函数是 x² + 5x。对外层求导得 3(x² + 5x)²，对内层求导得 2x + 5。因此：

dy/dx = 3(x² + 5x)² × (2x + 5)

Notice that we do not simplify the inner function when differentiating the outer layer. The inner expression (x² + 5x) remains exactly as it is in the first factor. Then we multiply by its derivative 2x + 5.

注意，对外层求导时我们不会化简内层函数。内层表达式 (x² + 5

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