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Mastering AQA A-Level Maths Question 7: Differentiation, Tangents and Stationary Points | 攻克AQA A-Level数学第7题：微分、切线与驻点

2026-08-21 19:51 · 本文部分内容由AI辅助创作

📚 Mastering AQA A-Level Maths Question 7: Differentiation, Tangents and Stationary Points | 攻克AQA A-Level数学第7题：微分、切线与驻点

In AQA A-Level Mathematics, Question 7 on Paper 1 often tests your ability to link differentiation with coordinate geometry, tangents, normals and stationary points. This article gives you a focused revision route through the exact skills examiners expect.

在AQA A-Level数学卷一中，第7题通常考查微分与坐标几何、切线、法线和驻点的综合应用。本文为你提供一条精准的复习路径，覆盖考官期望的关键技能。

1. Why Question 7 Feels Different | 为什么第7题感觉不一样

Question 7 in AQA Pure Maths Paper 1 is rarely a short 'differentiate this' item. Examiners typically embed differentiation inside a geometry or optimisation task, so you must interpret your derivative rather than stop after one line.

在AQA纯数学卷一里，第7题很少只是简短地“求导数”。考官通常把微分嵌入几何或优化任务中，因此你必须解释导数的意义，而不是写一行就停。

Common task types include finding the tangent or normal at a point, locating stationary points, classifying maxima and minima, and solving a real-world maximum or minimum problem.

常见题型包括：求某点的切线或法线、确定驻点、判断极大值与极小值，以及求解现实中的最大或最小问题。

2. Core Differentiation Rules You Must Recall | 必须掌握的核心微分法则

You will need these rules instantly. Write them on a revision card and test yourself both ways.

你需要立刻写出以下法则。把它们写在复习卡片上，并正反两个方向自测。

These building blocks are enough for most AQA Question 7 items once you add the chain, product and quotient rules for more complicated functions.

一旦加入链式法则、乘积法则和商法则，这些基本构件足以应对AQA第7题中的大多数函数。

d/dx (xⁿ) = n xⁿ⁻¹

d/dx (eˣ) = eˣ, d/dx (sin x) = cos x

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3. Finding Tangents and Normals | 求切线与法线

For a curve y = f(x), the gradient of the tangent at x = a is f' so find the y-coordinate f(a) before writing the equation, because the point is essential.

对于曲线 y = f(x)，在 x = a 处切线的斜率是 f'(a)。在写方程前，你还必须求出 y 坐标 f(a)，因为该点至关重要。

The tangent equation uses the straight-line formula with the point (a, f(a)) and gradient f'(a). The normal is perpendicular to the tangent, so its gradient is the negative reciprocal.

切线方程使用点斜式，点为 (a, f(a))，斜率为 f'(a)。法线与切线垂直，因此其斜率为负倒数。

Tangent: y - y₁ = m (x - x₁)

Normal: y - y₁ = -1/m (x - x₁)

Always use the negative reciprocal; a common error is to take the tangent gradient as the normal gradient by mistake.

始终使用负倒数；一个常见错误是误把切线斜率当作法线斜率。

4. Stationary Points: The Three Types | 驻点：三种类型

e tangent is horizontal, but the curve can turn up, turn down, or flatten out temporarily.

驻点出现在 dy/dx = 0 处。在这些点上切线是水平的，但曲线可能向上弯、向下弯或暂时平坦。