📚 AQA A-Level Maths Worked Example 7.1.1: Differentiation from First Principles | AQA A-Level 数学示例 7.1.1：从第一性原理求导
derstanding this limit-based method is essential for AQA Pure topics such as gradients of curves and tangent equations.
本例题精讲 AQA A-Level 数学中的示例 7.1.1，通常通过求 f(x) = x² 的导数引入从第一性原理进行微分。理解这种基于极限的方法对于 AQA Pure 中曲线梯度与切线方程等主题至关重要。
1. What is Example 7.1.1? | 什么是示例 7.1.1？
function of y = x² directly from the limit definition, rather than using a shortcut rule.
示例 7.1.1 是许多 AQA A-Level 数学课程微分章节中的开篇例题。它要求你直接从极限定义出发求 y = x² 的梯度函数，而不是使用快捷法则。
In AQA terms, this addresses assessment objective AO1 for accurate recall and AO2 for mathematical argument. The method becomes a foundation for later work on tangents, normals and optimisation.
在 AQA 考核中，这涉及准确记忆的 AO1 和数学论证的 AO2 目标。该方法为后续的切线、法线和优化问题打下基础。
2. AQA Specification Context | AQA 考试大纲背景
AQA A-Level Mathematics requires students to understand differentiation from first principles for simple functions such as x² and x³. The derivative is interpreted as the limit of the gradient of a chord as the chord approaches a tangent.
AQA A-Level 数学要求学生理解 x²、x³ 等简单函数从第一性原理求导。导数被理解为弦的斜率在弦趋向切线时的极限。
Questions may ask for a formal limit calculation, a geometrical interpretation, or the equation of a tangent at a given point. AQA often tests the difference quotient explicitly, so you must show every step of algebraic simplification.
题目可能要求正式的极限计算、几何解释或在给定点处的切线方程。AQA 经常明确考查差商，因此你必须展示代数化简的每一步。
3. The First-Principles Definition | 第一性原理定义
For a function f(x), the derivative f'(x) is defined as follows:
对于函数 f(x)，导数 f'(x) 定义如下：
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
This expression measures the rate at which f(x) changes over an interval h, then shrinks h to zero. The fraction inside the limit is called the difference quotient.
该表达式测量 f(x) 在区间 h 上的变化率，然后让 h 趋于零。极限内的分式称为差商。
Geometrically, [f(x + h) − f(x)] / h is the gradient of the chord joining (x, f(x)) and (x + h, f(x + h)). As h tends to 0, this chord turns into the tangent at x.
从几何角度看，[f(x + h) − f(x)] / h 是连接 (x, f(x)) 与 (x + h, f(x + h)) 两点的弦的斜率。当 h 趋于 0 时，这条弦变成点 x 处的切线。
4. Setting Up the Difference Quotient | 建立差商
Let f(x) = x². Replace x by x + h to obtain f(x + h) = (x + h)².
设 f(x) = x²。将 x 替换为 x + h，得到 f(x + h) = (x + h)²。
The difference quotient is therefore:
因此差商为：
[f(x + h) − f(x)] / h = [(x + h)² − x²] / h
If you do, both numerator and denominator become zero, giving the indeterminate form 0/0.
在这一阶段不要代入 h = 0。如果代入，分子和分母都会变成零，得到不确定形式 0/0。
5. Expanding and Simplifying | 展开并化简
Expand (x + h)² = x² + 2xh + h², then cancel the x² terms.
展开 (x + h)² = x² + 2xh + h²，然后消去 x² 项。
[(x² + 2xh + h²) − x²] / h = (2xh + h²) / h
Factor h from the numerator:
从分子中提取公因数 h：
(2xh + h²) / h = h(2x + h) / h = 2x + h, for h ≠ 0
We may cancel h because h is not zero during the limiting process. The limit examines values of h that are arbitrarily close to 0, but never equal to 0.
因为 h 在极限过程中不为零，所以可以约去 h。极限考查的是任意接近 0 但永远不等于 0 的 h 值。
6. Taking the Limit | 取极限
Apply the limit as h approaches zero:
令 h 趋近于 0，取极限：
f'(x) = lim(h→0) (2x + h) = 2x
0, the term 2x remains unchanged and the h term vanishes. Therefore, if f(x) = x², then f'(x) = 2x.
由于 h 趋于 0，2x 项保持不变，h 项消失。因此，若 f(x) = x²，则 f'(x) = 2x。
the gradient of the curve y = x² at any chosen x value, without needing to draw a tangent.
这最后一行就是导函数。它给出了曲线 y = x² 在任意选定 x 值处的梯度，无需绘制切线。
7. Interpreting the Result Geometrically | 几何解释结果
The derivative f'(x) = 2x gives the gradient of the curve y = x² at any point. At x = 3, the gradient is f'(3) = 6.
导数 f'(x) = 2x 给出了曲线 y = x² 在任意点处的梯度。在 x = 3 处，梯度为 f'(3) = 6。
Since y = 3² = 9, the tangent at (3, 9) has equation:
由于 y = 3² = 9，点 (3, 9) 处的切线方程为：
y − 9 = 6(x − 3) → y = 6x − 9
This geometric link between derivative and tangent slope is a recurring AQA exam theme. You should be able to move between f'(x), the gradient at a point, and the tangent equation quickly.
导数与切线斜率之间的几何联系是 AQA 考试中反复出现的主题。你应当能够快速在 f'(x)、某点处的梯度与切线方程之间进行转换。
8. Generalising to the Power Rule | 推广到幂函数法则
The same first-principles method can be used to show that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹ for rational n.
同样的第一性原理方法可以证明：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹，其中 n 为有理数。
For example, for f(x) = x³, expansion gives f(x + h) = x³ + 3x²h + 3xh² + h³, leading to f'(x) = 3x².
例如，对于 f(x) = x³，展开得到 f(x + h) = x³ + 3x²h + 3xh² + h³，从而 f'(x) = 3x²。
AQA candidates should recognise this pattern but must still be able to show the first-principles working when asked. A quick summary is shown below.
AQA 考生应识别这一规律，但在题目要求时仍必须展示第一性原理的推导过程。下面给出一个简要总结。
| f(x) | f'(x) |
|---|---|
| x² | 2x |
| x³ | 3x² |
| x⁴ | 4x³ |
The power rule is a valid shortcut in later work, but Example 7.1.1 shows where the rule comes from.
幂函数法则是后续学习中的有效快捷方法，但示例 7.1.1 展示了该法则的来源。
9. Common Mistakes | 常见错误
ore cancelling, producing a 0/0 expression. Always simplify the difference quotient first.
一个常见错误是在约分前就让 h = 0，从而产生 0/0 表达式。务必先化简差商。
Another error is forgetting that h ≠ 0 during cancellation, which justifies dividing by h. You should state this condition in your working.
另一个错误是忘记在约分过程中 h ≠ 0，这正是可以除以 h 的理由。你应该在解题过程中说明这一条件。
Students also sometimes expand (x + h)² incorrectly as x² + h². The correct expansion includes the cross term 2xh.
学生有时还会把 (x + h)² 错误地展开为 x² + h²。正确的展开应包含交叉项 2xh。
Do not set h = 0 too early.
不要过早令 h = 0。
Do not omit the condition h ≠ 0 when cancelling.
约分时不要遗漏 h ≠ 0 的条件。
Do not square a binomial incorrectly.
不要错误地平方二项式。
10. Exam-Style Practice | 考试风格练习
Try this AQA-style question: Given f(x) = x², use first principles to find f'(x) and hence the gradient of the curve at x = −2.
试试这道 AQA 风格题目：已知 f(x) = x²，用第一性原理求 f'(x)，并因此求出曲线在 x = −2 处的梯度。
Solution outline: f'(x) = 2x, so f'(−2) = −4. The tangent at (−2, 4) has gradient −4.
答案要点：f'(x) = 2x，所以 f'(−2) = −4。点 (−2, 4) 处的切线梯度为 −4。
Now extend it: find the equation of the tangent to y = x² at x = −2.
现在扩展：求曲线 y = x² 在 x = −2 处的切线方程。
y − 4 = −4(x + 2) → y = −4x − 4
Mark schemes often award method marks for the limit setup, expansion, cancellation, and final interpretation, even if a sign error appears later.
评分方案通常会为极限式的建立、展开、约分以及最终解释授予方法分，即使后续出现符号错误也是如此。
11. Why This Example Matters | 为什么这个示例很重要
orces you to understand the limiting process behind differentiation. Once you can derive f' x from first principles, you are less likely to misuse rules in more complex AQA problems.
示例 7.1.1 很重要，因为它促使你理解微分背后的极限过程。一旦你能从第一性原理推导出 f'(x) = 2x，在更复杂的 AQA 问题中就不太可能误用规则。
It also builds the vocabulary AQA examiners expect: gradient of a chord, limit as h tends to 0, and tangent at a point.
它也建立了 AQA 阅卷人所期望的词汇：弦的斜率、h 趋于 0 的极限，以及某点处的切线。
12. Summary | 总结
mysterious rule but a limit of average rates of change. For f(x) = x², the derivative is f'(x) = 2x.
示例 7.1.1 表明，微分并不是神秘的规则，而是平均变化率的极限。对于 f(x) = x²，其导数为 f'(x) = 2x。
Mastering this method builds a secure foundation for tangents, increasing and decreasing functions, and optimisation problems in AQA A-Level Mathematics.
掌握这种方法为 AQA A-Level 数学中的切线、增函数与减函数以及优化问题打下坚实基础。
In an exam, always show the three core steps: write the difference quotient, simplify algebraically, and then take
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