📚 Example 4.2.1: Differentiation from First Principles | 示例 4.2.1：从第一原理求导
In this worked example, we explore one of the most fundamental ideas in AQA A Level Mathematics: differentiation from first principles. Example 4.2.1 shows how the derivative of a simple polynomial function can be obtained directly from the limit definition, without relying on shortcut rules.
在这个示例中，我们探讨 AQA A Level 数学中最基本的思想之一：从第一原理求导。示例 4.2.1 展示了如何直接从极限定义出发求一个简单多项式函数的导数，而不依赖任何捷径法则。
1. The First Principle Definition | 第一原理定义
For a function f(x), the derivative at a general point x is defined as f' ded the limit exists. This expression measures the instantaneous rate of change of f(x) with respect to x.
对于函数 f(x)，其在一般点 x 处的导数定义为 f'(x) = limₕ→₀ [f(x + h) − f(x)] / h，前提是该极限存在。这个表达式度量了 f(x) 关于 x 的瞬时变化率。
f'(x) = limₕ→₀ [f(x + h) − f(x)] / h
2. Understanding the Limit Notation | 理解极限符号
increment h tend to zero. We do not set h equal to zero immediately, because that would give 0/0, which is undefined. Instead, we simplify the quotient first and then examine what value it approaches.
符号 limₕ→₀ 表示我们让一个微小增量 h 趋近于零。我们不能立即将 h 设为零，因为那样会得到 0/0，这是未定义的。相反，我们首先化简这个商，然后观察它趋近于什么值。
3. Step 1: Write the Difference Quotient | 步骤一：写出差商
To differentiate f(x) from first principles, start by calculating f(x + h). Replace every x in the original expression with x + h. Then write the difference quotient [f(x + h) − f(x)] / h.
要从第一原理对 f(x) 求导，首先计算 f(x + h)。将原表达式中的每一个 x 替换为 x + h。然后写出差商 [f(x + h) − f(x)] / h。
- Find f(x + h) by substitution.
- Subtract the original function f(x).
- Divide the result by the small change h.
4. Step 2: Expand and Simplify | 步骤二：展开并化简
After substituting, expand brackets and collect like terms. The goal is to remove the fraction so that the h in the denominator can be cancelled. For polynomial functions, this usually produces terms containing at least one factor of h in the numerator.
代入后，展开括号并合并同类项。目的是通过化简消去分母中的 h。对于多项式函数，这通常会使分子中的每一项都至少含有一个因子 h。
5. Step 3: Cancel h and Take the Limit | 步骤三：约去 h 并取极限
Once the numerator has a common factor h, cancel it with the denominator. Then let h tend to zero in the remaining expression. Any term still containing h will vanish, leaving the derivative f'(x).
一旦分子有公因子 h，就将其与分母约去。然后令 h 在剩余表达式中趋近于零。任何仍含有 h 的项都会消失，留下导数 f'(x)。
6. Worked Example: f(x) = x² | 例题：f(x) = x²
h²) − x²] / h = (2xh + h²) / h = h(2x + h) / h = 2x + h. Taking the limit as h → 0 gives f'(x) = 2x.
设 f(x) = x²。则 f(x + h) = (x + h)² = x² + 2xh + h²。差商变为 [(x² + 2xh + h²) − x²] / h = (2xh + h²) / h = h(2x + h) / h = 2x + h。令 h → 0 取极限，得到 f'(x) = 2x。
f'(x) = limₕ→₀ [(x + h)² − x²] / h = limₕ→₀ (2x + h) = 2x
7. Worked Example: f(x) = 3x² − 5x | 例题：f(x) = 3x² − 5x
. Divide by h to obtain (6xh + 3h² − 5h) / h = 6x + 3h − 5. Letting h tend to zero gives f'(x) = 6x − 5.
设 f(x) = 3x² − 5x。首先，f(x + h) = 3(x + h)² − 5(x + h) = 3(x² + 2xh + h²) − 5x − 5h = 3x² + 6xh + 3h² − 5x − 5h。减去 f(x) 得到 6xh + 3h² − 5h。除以 h 得到 (6xh + 3h² − 5h) / h = 6x + 3h − 5。令 h 趋近于零，得到 f'(x) = 6x − 5。
8. Alternative Notations for the Derivative | 导数的其他记号
ve can be written in several equivalent ways. The most common notations for y = f(x) are f'(x), dy/dx, y', and d/dx [f(x)]. All represent the same limit and the same gradient function.
导数可以用几种等价的方式表示。对于 y = f(x)，最常见的记号有 f'(x)、dy/dx、y' 和 d/dx [f(x)]。它们都表示同一个极限和同一个梯度函数。
- f'(x) — Lagrange notation
- dy/dx — Leibniz notation
- y' — Newton notation (less common at A level)
- d/dx [f(x)] — operator notation
9. Common Mistakes to Avoid | 常见错误提醒
producing 0/0. Another mistake is forgetting to subtract f(x) completely, especially when f(x) contains negative terms. Also, do not expand (x + h)² as x² + h²; the correct expansion includes the cross term 2xh.
一个常见错误是过早地令 h = 0，从而得到 0/0。另一个错误是忘记完全减去 f(x)，尤其是当 f(x) 含有负项时。此外，不要将 (x + h)² 展开为 x² + h²；正确的展开包含交叉项 2xh。
- Never replace h with 0 before cancelling.
- Always subtract the whole of f(x), not just part of it.
- Expand brackets fully and check for cross terms.
10. Linking to Tangent Gradients | 与切线斜率的联系
The derivative f' of the tangent to the curve y = f(x) at any point x = a. For example, if f(x) = x², then f' x = 2 has gradient 4. This geometric interpretation is central to many AQA exam questions.
导数 f'(x) 给出了曲线 y = f(x) 在任意点 x = a 处切线的斜率。例如，若 f(x) = x²，则 f'(2) = 4，因此 x = 2 处的切线斜率为 4。这种几何解释是许多 AQA 考题的核心。
11. Exam Tips for AQA | AQA 考试技巧
In AQA exams, you may be asked to differentiate a simple polynomial from first principles as a proof-style question. Show every line of working, including the limit notation. Even if you know the power rule, you must use the first principle definition when the question explicitly requires it.
在 AQA 考试中，你可能会被要求以证明题的形式从第一原理求一个简单多项式的导数。展示每一步计算，包括极限符号。即使你知道幂法则，当题目明确要求时，也必须使用第一原理定义。
- Write f(x + h) explicitly before simplifying.
- Keep the limit notation until after cancellation.
- State the final derivative clearly as f'(x).
12. Practice Question | 练习题
Using first principles, find the derivative of f(x) = x² + 4x. Check that your answer is f' Write out f(x + h), form the difference quotient, simplify, cancel h, and take the limit.
试试这个问题：用第一原理求 f(x) = x² + 4x 的导数。检查你的答案是否为 f'(x) = 2x + 4。写出 f(x + h)，构造差商，化简，约去 h，并取极限。
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