AQA Maths: Differentiation – Key Points Explained | AQA 数学：微分考点精讲

📚 AQA Maths: Differentiation – Key Points Explained | AQA 数学：微分考点精讲

Differentiation is one of the core topics in AQA A-Level Mathematics. It forms the foundation of calculus and allows us to study instantaneous rates of change, find gradients of curves, and solve optimisation problems. Mastering the rules of differentiation and understanding their geometric meaning are essential for success in the exam. This article walks you through the key concepts and techniques you need to know, with clear explanations and practical tips.

微分是 AQA A-Level 数学的核心主题之一。它是微积分学的基础，能够帮助我们研究瞬时变化率、求解曲线梯度以及解决最优化问题。掌握求导法则并理解其几何意义，是考试成功的关键。本文将带你逐一梳理必须掌握的重要概念和技巧，配以清晰的解释和实用提示。

1. What is Differentiation? | 什么是微分？

Differentiation is the process of finding the derivative of a function. The derivative, denoted by dy/dx or f'(x), measures the rate at which the dependent variable y changes with respect to the independent variable x. Geometrically, it gives the gradient of the tangent to the curve at any point. The formal definition involves limits, but in practice we rely on derivative rules.

微分是求函数导数的过程。导数记作 dy/dx 或 f'(x)，它测量了因变量 y 关于自变量 x 的变化率。从几何上看，导数给出了曲线上任意一点处切线的斜率。形式化定义涉及极限，但在实际运算中我们依赖求导法则。

2. Basic Derivative Rules | 基本求导法则

For a constant function y = c, the derivative is zero: d/dx (c) = 0.

对于常数函数 y = c，导数为零：d/dx (c) = 0。

The power rule is the most commonly used rule. If y = xⁿ, then dy/dx = n xⁿ⁻¹. Recall that x can be written as x¹, and 1/x as x⁻¹. This rule works for any real exponent n.

幂法则是使用最频繁的法则。若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。注意 x 可写作 x¹，1/x 可写作 x⁻¹。该法则适用于任意实数指数 n。

For y = k·f(x), where k is a constant, dy/dx = k·f'(x). This constant multiple rule lets you pull constants out of the differentiation.

对于 y = k·f(x)，其中 k 为常数，dy/dx = k·f'(x)。这一常数倍法则允许你将常数提到微分号外面。

3. Sum and Difference Rule | 和差法则

The derivative of a sum or difference of functions is the sum or difference of their derivatives. If y = f(x) ± g(x), then dy/dx = f'(x) ± g'(x). This rule allows you to differentiate term by term, which is absolutely fundamental when handling polynomial functions in AQA exams.

函数和或差的导数等于它们各自导数的和或差。若 y = f(x) ± g(x)，则 dy/dx = f'(x) ± g'(x)。这一法则允许你逐项求导，在处理多项式函数时是 AQA 考试中最基本的方法。

4. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数

The exponential function eˣ is unique because its derivative is itself: d/dx (eˣ) = eˣ. More generally, for y = eᵏˣ, the chain rule gives dy/dx = k eᵏˣ. For the natural logarithm, d/dx (ln x) = 1/x for x > 0.

指数函数 eˣ 很特别，因为它的导数就是它本身：d/dx (eˣ) = eˣ。更一般地，对于 y = eᵏˣ，利用链式法则可得 dy/dx = k eᵏˣ。对于自然对数，d/dx (ln x) = 1/x（x > 0）。

5. Derivatives of Trigonometric Functions | 三角函数的导数

Trigonometric derivatives appear frequently in AQA papers. The key results are: d/dx (sin x) = cos x; d/dx (cos x) = – sin x; and d/dx (tan x) = sec² x. Remember that these are only valid when x is measured in radians. Always check the angle unit in the question.

三角函数的导数在 AQA 试卷中出现频率很高。核心结论是：d/dx (sin x) = cos x；d/dx (cos x) = – sin x；d/dx (tan x) = sec² x。切记这些公式仅在 x 采用弧度制时成立。作答时务必检查题目中的角度单位。

6. Product Rule | 积法则

When you differentiate the product of two functions, y = u v, where both u and v are functions of x, you must apply the product rule: dy/dx = u dv/dx + v du/dx. A typical mistake is to simply multiply the derivatives. In AQA exams, look for expressions like x sin x or eˣ ln x.

当你对两个函数的乘积求导，y = u v，其中 u 和 v 都是 x 的函数，必须使用积法则：dy/dx = u dv/dx + v du/dx。常见的错误是直接将导数相乘。在 AQA 考试中，留意诸如 x sin x 或 eˣ ln x 的表达式。

7. Quotient Rule | 商法则

For a quotient of two functions, y = u / v, the derivative is given by the quotient rule: dy/dx = (v du/dx – u dv/dx) / v². The order matters: you must differentiate the numerator first with a minus sign. Use this rule for rational functions like (x² + 1)/(x – 3).

对于两个函数的商 y = u / v，导数由商法则给出：dy/dx = (v du/dx – u dv/dx) / v²。注意运算顺序：必须先处理分子的导数，并使用减号。对形如 (x² + 1)/(x – 3) 的有理函数求导时，就要用到这个法则。

8. Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). A handy way to remember it is: dy/dx = dy/du × du/dx, where u = g(x). This rule is crucial for dealing with powers like (ax + b)ⁿ, exponentials such as e²ˣ, and trigonometric composites like sin(3x).

链式法则用于求复合函数的导数。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。一个便于记忆的方法是：dy/dx = dy/du × du/dx，其中 u = g(x)。该法则在处理形如 (ax + b)ⁿ 的幂函数、e²ˣ 这样的指数函数以及 sin(3x) 等三角复合函数时至关紧要。

9. Gradient, Tangents and Normals | 梯度、切线方程与法线方程

Once you have the derivative dy/dx, you can find the gradient of a curve at a specific point by substituting the x-coordinate. The equation of the tangent line at (x₁, y₁) is y – y₁ = m(x – x₁), where m = dy/dx evaluated at x₁. The normal line is perpendicular to the tangent, so its gradient is –1/m. These geometric applications are very common in AQA exam questions.

一旦得到导数 dy/dx，你就可以代入 x 坐标求出曲线在某一点处的梯度。在点 (x₁, y₁) 处的切线方程为 y – y₁ = m(x – x₁)，其中 m 是 dy/dx 在 x₁ 处的值。法线与切线垂直，因此其斜率为 –1/m。这些几何应用在 AQA 试题中十分常见。

10. Increasing/Decreasing Functions and Stationary Points | 增减性与驻点

A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. A stationary point occurs where f'(x) = 0. There are three types: local maximum, local minimum, and point of inflection. The second derivative f''(x) helps classify them: if f''(x) < 0 at a stationary point, it is a maximum; if f''(x) > 0, it is a minimum. When f”(x) = 0, further investigation is needed.

在 f'(x) > 0 的区域函数递增，在 f'(x) < 0 的区域函数递减。驻点出现在 f'(x) = 0 处。驻点有三种类型：局部极大值、局部极小值和拐点。二阶导数 f''(x) 有助于进行分类：若在驻点处 f''(x) < 0，则为极大值点；若 f''(x) > 0，则为极小值点。当 f”(x) = 0 时，需要进一步分析。

11. Real‑World Rates of Change | 实际变化率应用

In AQA exams, differentiation is often tested in context, such as modelling motion, growth, or geometrical optimisation. The derivative represents the instantaneous rate of change: for displacement s(t), ds/dt gives velocity, and d²s/dt² gives acceleration. For optimisation problems, setting dA/dx = 0 or dV/dx = 0 gives the maximum or minimum value of an area or volume.

在 AQA 考试中，微分常常被放在实际情境中考查，例如运动建模、增长模型或几何最优化。导数表示瞬时变化率：对于位移 s(t)，ds/dt 表示速度，d²s/dt² 表示加速度。在最优化问题中，令 dA/dx = 0 或 dV/dx = 0 便可求得面积或体积的最大值或最小值。

12. AQA Exam Tips | AQA 考试技巧

Always simplify your answer after differentiating, and present it in the simplest form possible. Watch out for hidden factors when using the chain rule, and remember to multiply by the derivative of the inner function. For quotient rule, double-check that the denominator is squared. In applied problems, clearly state the stationary point you found and verify with the second derivative. Finally, use radians in calculus unless the question specifies degrees.

求导后始终要化简答案，并尽可能将其整理成最简形式。使用链式法则时要警惕隐藏的因子，牢记乘以内层函数的导数。对于商法则，反复检查分母是否平方。在应用题中，清晰说明你所找到的驻点，并用二阶导数进行验证。最后，除非题目明确说明角度单位，否则微积分部分全部使用弧度制。

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AQA Maths: Differentiation – Key Points Explained | AQA 数学：微分 考点精讲