📚 AQA A-Level Maths Example 1.5.1: Differentiation from First Principles | AQA A-Level 数学示例 1.5.1：从第一性原理求导
o show that differentiation is not just a rule to memorise but a limiting process that measures instantaneous rate of change.
在这个 AQA A-Level 数学例题中，我们研究如何直接从极限定义求出简单函数的导数。示例 1.5.1 使用函数 f(x) = x² 来表明，求导不只是需要记忆的规则，而是一种测量瞬时变化率的极限过程。
1. What Does 'First Principles' Mean? | “第一性原理”是什么意思？
Differentiation from first principles means finding the gradient of a curve by taking the limit of the gradient of a chord joining two nearby points. No standard differentiation rules such as the power rule are assumed at the start.
从第一性原理求导意味着通过取连接两个相邻点的弦的斜率极限来求曲线的斜率。开始时并不假设使用幂法则等标准求导规则。
The idea is central to AQA papers because it tests whether you understand the definition of the derivative, not just whether you can apply it. In Example 1.5.1 we prove a familiar result using only algebraic manipulation and a limit.
这一思想是 AQA 试卷的核心，因为它考查你是否理解导数的定义，而不仅仅是你是否会应用它。在示例 1.5.1 中，我们仅通过代数运算和一个极限来证明一个熟悉的结果。
2. The Limit Definition of the Derivative | 导数的极限定义
For a function f(x), the derivative f'(x) is defined by the following limit, provided the limit exists:
对于函数 f(x)，导数 f'(x) 由以下极限定义，前提是该极限存在：
f'(x) = lim (h → 0) [ f(x + h) − f(x) ] / h
Here h represents a small change in x. The expression [f(x + h) − f(x)] / h is called the difference quotient. It gives the average rate of change between x and x + h.
这里 h 表示 x 的一个微小变化。表达式 [f(x + h) − f(x)] / h 称为差商。它给出了 x 与 x + h 之间的平均变化率。
at the point x. This limiting value is the gradient of the tangent to the curve y = f(x).
当 h 趋近于零时，平均变化率趋近于 x 处的瞬时变化率。这个极限值就是曲线 y = f(x) 的切线斜率。
Some textbooks use the symbol δx instead of h. The meaning is identical: δx is a small increment in x that eventually tends to zero.
有些教材使用符号 δx 代替 h。其含义完全相同：δx 是 x 的一个小增量，最终趋近于零。
3. Setting Up the Difference Quotient for f(x) = x² | 为 f(x) = x² 构造差商
.1, we take f(x) = x². First, evaluate f(x + h) by replacing x with x + h in the function.
在示例 1.5.1 中，我们取 f(x) = x²。首先，将函数中的 x 替换为 x + h，求出 f(x + h)。
f(x + h) = (x + h)²
Then the difference quotient becomes:
因此差商变为：
[ (x + h)² − x² ] / h
It is essential to use brackets around the numerator because the whole expression f(x + h) − f(x) is divided by h. If the brackets are omitted, the notation becomes ambiguous and often leads to algebra errors.
在分子周围使用括号非常重要，因为整个表达式 f(x + h) − f(x) 都要除以 h。如果省略括号，符号就会变得含糊不清，并经常导致代数错误。
At this stage, you should not let h equal zero. The expression is an average gradient over a nonzero interval, so h is strictly positive or negative but never zero.
在这一阶段，不应该让 h 等于零。该表达式是非零区间上的平均斜率，因此 h 严格为正或为负，但绝不为零。
4. Expanding and Simplifying | 展开与化简
Expand (x + h)² carefully using the binomial expansion or the distributive law:
使用二项式展开或分配律仔细展开 (x + h)²：
(x + h)² = x² + 2xh + h²
Substitute this into the difference quotient:
将其代入差商：
[ x² + 2xh + h² − x² ] / h
The x² and −x² cancel, leaving a numerator that has a common factor of h.
x² 与 −x² 相互抵消，留下的分子含有公因子 h。
[ 2xh + h² ] / h
As long as h is not zero, we can simplify by dividing each term by h:
只要 h 不等于零，我们就可以将每一项除以 h 来化简：
2x + h
This simplification is valid before taking the limit because h is only approaching zero, not equal to zero at this stage. It is the key algebraic step that removes the zero denominator.
这种化简在取极限之前是有效的，因为在这一阶段 h 只是趋近于零，并不等于零。这是消除零分母的关键代数步骤。
5. Taking the Limit as h → 0 | 令 h → 0 取极限
Now apply the limit to the simplified expression 2x + h. Since h tends to zero, the term h vanishes.
现在对化简后的表达式 2x + h 取极限。由于 h 趋近于零，h 项消失。
f'(x) = lim (h → 0) (2x + h) = 2x
hes the power rule result d/dx(x²) = 2x, but here it has been derived from the definition.
因此，x² 对 x 的导数为 2x。这与幂法则的结果 d/dx(x²) = 2x 一致，但这里是从定义推导出来的。
ituting h = 0. For example, write lim (h → 0) (2x + h) and then state that this equals 2x.
在考试答案中，你应在代入 h = 0 之前清楚地展示极限符号。例如，写出 lim (h → 0) (2x + h)，然后说明它等于 2x。
6. Interpreting the Result | 解释结果
The derivative f' lf a function. It tells us the gradient of the tangent to the curve y = x² at any point x.
导数 f'(x) = 2x 本身也是一个函数。它告诉我们曲线 y = x² 在任意点 x 处切线的斜率。
xample, at x = 3, the gradient of the tangent is 2 × 3 = 6. At x = −2, the gradient is −4.
例如，在 x = 3 处，切线斜率为 2 × 3 = 6。在 x = −2 处，斜率为 −4。
| x | f'(x)
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