Example 6.4.1: Differentiating Composite Functions with the Chain Rule | 例题6.4.1：用链式法则求复合函数的导数

📚 Example 6.4.1: Differentiating Composite Functions with the Chain Rule | 例题6.4.1：用链式法则求复合函数的导数

Welcome to this step-by-step guide to a classic AQA A-Level Mathematics exercise, Example 6.4.1. In this problem, we are asked to differentiate a composite function using the chain rule. The chain rule is a fundamental technique in calculus, and it is regularly tested in AQA Pure Mathematics papers. By the end of this article, you will not only understand how to solve this exact example, but also be able to apply the same method to a wide range of similar questions.

欢迎阅读AQA A-Level数学经典例题6.4.1的分步解析。本题需要我们运用链式法则对复合函数进行求导。链式法则是微积分中的基础方法，在AQA纯数学考卷中考查频率极高。学完这篇文章，你不仅能够掌握这一例题的解法，还能将同样的方法灵活运用于各类类似的题目。

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1. The Problem | 题目

Consider the function:

考虑函数：

y = (2x³ − 5x)⁶

Find d y/d x.

求 d y/d x。

This expression is a composite function: the outer part is the sixth power, while the inner part is the polynomial 2x³ − 5x. Our task is to differentiate it correctly.

该表达式是一个复合函数：外层是六次幂，内层是多项式 2x³ − 5x。我们的任务就是要正确求导。

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2. The Chain Rule Formula | 链式法则公式

The chain rule states that if y is a function of u, and u is a function of x, then:

链式法则指出，若 y 是 u 的函数，且 u 是 x 的函数，则：

d y/d x = d y/d u × d u/d x

This rule allows us to break down the derivative of a composite function into two simpler derivatives. It is often summarised as: differentiate the outer function, multiply by the derivative of the inner function.

这条法则让我们把复合函数的导数拆成两个更简单的导数。其核心口诀是：外层函数求导，再乘以内层函数的导数。

In this example, we can set u = 2x³ − 5x, so that y = u⁶. Then d y/d x can be found by multiplying d y/d u and d u/d x.

在本题中，我们令 u = 2x³ − 5x，则 y = u⁶。于是 d y/d x 可以通过 d y/d u 与 d u/d x 相乘得到。

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3. Identifying the Inner and Outer Functions | 识别内外函数

Before applying the chain rule, it is vital to identify which part is the inner function and which is the outer function.

在应用链式法则之前，准确判断内函数与外函数至关重要。

Here, the inner function is the expression inside the brackets:

本题中，括号内的表达式就是内函数：

u = 2x³ − 5x

The outer function is the operation applied to u, namely raising u to the power 6:

外函数是对 u 进行的操作，即取 u 的 6 次方：

y = u⁶

Note that the order matters. If we mistakenly treat the polynomial as the outer function, our derivative will be completely wrong.

注意内外函数的顺序不能颠倒。如果我们错误地把多项式当作外函数，求导结果将完全错误。

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4. Differentiating the Outer Function | 对外函数求导

Now we differentiate the outer function y = u⁶ with respect to u. Using the power rule:

现在我们对关于 u 的外函数 y = u⁶ 求导。根据幂函数求导法则：

d y/d u = 6u⁵

We bring the index 6 down in front, and reduce the power by 1. This gives 6u⁵.

将指数 6 移到前面，指数减 1，于是得到 6u⁵。

Notice that we do not yet differentiate the inner part. This is a common source of error when students rush ahead.

注意此时我们尚未对内函数求导。这也是许多同学在匆忙中犯错的常见原因。

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5. Differentiating the Inner Function | 对内函数求导

Next, we differentiate the inner function u = 2x³ − 5x with respect to x.

接下来，我们关于 x 对内

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