Worked Example 7.2.1: Differentiation from First Principles | 例题 7.2.1：用第一原理求导

📚 Worked Example 7.2.1: Differentiation from First Principles | 例题 7.2.1：用第一原理求导

This worked example demonstrates how to use the definition of the derivative from first principles, a key skill in the AQA A-Level Mathematics Pure Maths specification. We will solve the classic problem of differentiating f(x) = x² and discuss the underlying ideas in a clear, step-by-step manner.

本例题演示如何运用第一原理定义求导数，这是AQA A-Level数学纯数学大纲中的一项关键技能。我们将逐步求解 f(x) = x² 的导数这个经典问题，并清晰地讨论其背后的思想。

1. The Question | 例题题目

Example 7.2.1 states: Use the definition of the derivative from first principles to find f′(x) for f(x) = x².

例题7.2.1：使用第一原理中导数的定义，求函数 f(x) = x² 的导数 f′(x)。

This may seem simple because the power rule gives the answer instantly. However, the aim of this exercise is to prove that rule for this special case using the limit definition.

这道题看似简单，因为幂法则可以立即给出答案。然而，这个练习的目的是用极限定义来证明这一法则在该特殊情况下的成立。

2. Why Use First Principles? | 为什么要用第一原理？

The derivative represents the instantaneous rate of change of a

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