Mastering Differentiation Rules for Edexcel A-Level Pure Maths | 精通 Edexcel A-Level 纯数学微分法则

📚 Mastering Differentiation Rules for Edexcel A-Level Pure Maths | 精通 Edexcel A-Level 纯数学微分法则

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. A strong command of first principles, the chain rule, the product rule and the quotient rule is essential for Pure Mathematics, mechanics and any applied context involving rates of change.

微分是 Edexcel A-Level 数学中考查最频繁的主题之一。熟练掌握第一原理、链式法则、乘法法则和除法法则对于纯数学、力学以及任何涉及变化率的应用问题都至关重要。

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1. Derivative as a Limit | 作为极限的导数

The derivative measures the instantaneous rate of change of a function at a given point, and it is defined formally using a limit.

导数衡量函数在某一点上的瞬时变化率，它通过极限来给出正式定义。

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

This first-principles definition underpins every differentiation rule used in Edexcel A-Level Mathematics. Students should be able to apply it to simple polynomials such as f(x) = x².

这一第一原理定义是 Edexcel A-Level 数学中所有微分法则的基础。学生应能够将其应用于 f(x) = x² 这类简单多项式。

For f(x) = x², expanding gives [(x+h)² − x²]/h = (2xh + h²)/h = 2x + h, so the limit as h tends to 0 is 2x.

对于 f(x) = x²，展开后得到 [(x+h)² − x²]/h = (2xh + h²)/h = 2x + h，因此当 h 趋向于 0 时极限为 2x。

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2. Power Rule and Basic Rules | 幂函数法则与基本法则

The power rule is the foundation of most A-Level differentiation work.

幂函数法则是大多数 A-Level 微分运算的基础。

If y = xⁿ, then dy/dx = n xⁿ⁻¹

This rule works for any real constant n, including negative and fractional powers. For example, y = x⁻² differentiates to dy/dx = −2x⁻³, and y = √x = x^½ differentiates to dy/dx = ½ x^−½.

该法则适用于任何实数常数 n，包括负指数和分数指数。例如，y = x⁻² 的导数为 dy/dx = −2x⁻³；y = √x = x^½ 的导数为 dy/dx = ½ x^−½。

• Constant rule: d/dx (c) = 0
• 常数法则：d/dx (c) = 0
• Constant multiple rule: d/dx [c f(x)] = c f'(x)
• 常数倍法则：d/dx [c f(x)] = c f'(x)
• Sum and difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
• 和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

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3. Chain Rule | 链式法则

The chain rule is used when one function is composed inside another. It is one of the most frequently required skills in Edexcel A-Level Pure Mathematics.

链式法则用于一个函数复合在另一个函数内部的情形。它是 Edexcel A-Level 纯数学中最常考查的技能之一。

dy/dx = dy/du × du/dx

If y = (3x² + 5)⁴, set u = 3x² + 5, then y = u⁴. Therefore dy/du = 4u³ and du/dx = 6x.

如果 y = (3x² + 5)⁴，设 u = 3x² + 5，则 y = u⁴。因此 dy/du = 4u³，du/dx = 6x。

dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³

The chain rule can be extended to multiple layers, but at A-Level most questions involve one or two substitutions. Always remember to multiply by the derivative of the inner function.

链式法则可以扩展到多层复合函数，但 A-Level 中的大多数问题只涉及一层或两层代换。一定要记得乘以内层函数的导数。

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4. Product Rule | 乘法法则

When differentiating the product of two functions, the product rule must be used.

当对两个函数的乘积求导时，必须使用乘法法则。

If y = u v, then dy/dx = u’ v + u v’

Here u and v are both functions of x, and the prime denotes differentiation with respect to x.

这里 u 和 v 都是 x 的函数，撇号表示对 x 求导。

Example: differentiate y = x² eˣ. Let u = x² and v = eˣ, so u’ = 2x and v’ = eˣ.

示例：求 y = x² eˣ 的导数。设 u = x²，v = eˣ，则 u’ = 2x，v’ = eˣ。

dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x)

In exam questions, always identify u and v clearly before applying the rule. Avoid the common mistake of simply multiplying the two individual derivatives together.

在考试题中，应用法则前应先清楚地确定 u 和 v。避免简单地将两个单独的导数相乘这一常见错误。

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5. Quotient Rule | 除法法则

The quotient rule is required when differentiating one function divided by another.

当对一个函数除以另一个函数的形式求导时，需要用到除法法则。

If y = u / v, then dy/dx = (v u’ − u v’) / v²

The order in the numerator is critical: v times the derivative of u, minus u times the derivative of v, all divided by v squared.

分子中的顺序非常关键：v 乘以 u 的导数，减去 u 乘以 v 的导数，再整体除以 v 的平方。

Example: differentiate y = (x² + 1) / (x − 1). Let u = x² + 1 and v = x − 1, so u’ = 2x and v’ = 1.

示例：求 y = (x² + 1) / (x − 1) 的导数。设 u = x² + 1，v = x − 1，则 u’ = 2x，v’ = 1。

dy/dx = [(x − 1)(2x) − (x² + 1)(1)] / (x − 1)² = (2x² − 2x − x² − 1) / (x − 1)² = (x² − 2x − 1) / (x − 1)²

Quotient rule questions often appear in A-Level exam papers with algebraic simplification required after differentiation. Take care when expanding and collecting like terms.

除法法则题目经常出现在 A-Level 试卷中，并需要在求导后进行代数化简。展开和合并同类项时要特别小心。

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6. Second Derivative and Stationary Points | 二阶导数与驻点

The second derivative f”(x) measures the rate of change of f'(x), and it helps classify stationary points.

二阶导数 f”(x) 衡量 f'(x) 的变化率，它有助于判断驻点的类型。

Stationary point: f'(x) = 0

To classify a stationary point, substitute its x-value into the second derivative.

要判断驻点类型，应将其 x 值代入二阶导数。

• If f”(x) > 0, the point is a local minimum.
• 如果 f”(x) > 0，该点为局部极小值。
• If f”(x) < 0, the point is a local maximum.
• 如果 f”(x) < 0，该点为局部极大值。
• If f”(x) = 0, further investigation is needed.
• 如果 f”(x) = 0，则需要进一步分析。

Example: f(x) = x³ − 3x. Then f'(x) = 3x² − 3, so solve 3x² − 3 = 0 to get x = ±1.

示例：f(x) = x³ − 3x。则 f'(x) = 3x² − 3，解 3x² − 3 = 0 得到 x = ±1。

Since f”(x) = 6x, f”(1) = 6 > 0 so x = 1 is a minimum, and f”(−1) = −6 < 0 so x = −1 is a maximum.

因为 f”(x) = 6x，f”(1) = 6 > 0，所以 x = 1 处为极小值；f”(−1) = −6 < 0，所以 x = −1 处为极大值。

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7. Tangents and Normals | 切线与法线

Differentiation is used to find the equation of a tangent or normal to a curve at a given point.

微分可用于求曲线在某给定点处的切线和法线方程。

Tangent: y − y₁ = m(x − x₁), where m = dy/dx

For a normal, the gradient is the negative reciprocal of the tangent gradient.

对于法线，其斜率为切线斜率的负倒数。

Normal gradient: m_normal = −1 / m_tangent

Example: find the tangent to y = x² at x = 3. Since dy/dx = 2x, the gradient at x = 3 is m = 6, and y = 9.

示例：求 y = x² 在 x = 3 处的切线。因为 dy/dx = 2x，x = 3 处斜率为 m = 6，此时 y = 9。

Tangent equation: y − 9 = 6(x − 3), or y = 6x − 9

For the normal at the same point, the gradient is −1/6, giving y − 9 = −1/6 (x − 3).

同一点处的法线斜率为 −1/6，因此方程为 y − 9 = −1/6 (x − 3)。

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8. Increasing and Decreasing Functions | 递增函数与递减函数

The sign of the first derivative determines whether a function is increasing or decreasing on an interval.

一阶导数的符号决定函数在某一区间上是递增还是递减。

• If f'(x) > 0 for all x in an interval, f is increasing on that interval.
• 如果在某区间上 f'(x) > 0，则 f 在该区间上递增。
• If f'(x) < 0 for all x in an interval, f is decreasing on that interval.
• 如果在某区间上 f'(x) < 0，则 f 在该区间上递减。

Example: for f(x) = x³ − 3x, we have f'(x) = 3(x − 1)(x + 1). A sign diagram shows f'(x) > 0 for x < −1 and x > 1, so f is increasing there.

示例：对于 f(x) = x³ − 3x，f'(x) = 3(x − 1)(x + 1)。符号图显示当 x < −1 和 x > 1 时 f'(x) > 0，因此 f 在这些区间上递增。

For −1 < x < 1, f'(x) < 0, so f is decreasing on (−1, 1).

当 −1 < x < 1 时，f'(x) < 0，因此 f 在区间 (−1, 1) 上递减。

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9. Connected Rates of Change | 相关变化率

Connected rates of change link two or more variables that change with respect to time. The chain rule is the key tool here.

相关变化率将两个或多个随时间变化的变量联系起来。链式法则是处理这类问题的关键工具。

dV/dt = dV/dr × dr/dt

Example: the radius r of a spherical balloon increases at 2 cm s⁻¹. Find the rate of change of volume when r = 5 cm.

示例：一个球形气球的半径以 2 cm s⁻¹ 的速度增大。求当 r = 5 cm 时体积的变化率。

The volume of a sphere is V = 4/3 π r³, so dV/dr = 4π r².

球体积为 V = 4/3 π r³，因此 dV/dr = 4π r²。

dV/dt = 4π r² × 2 = 8π r²

Substituting r = 5 gives dV/dt = 8π × 25 = 200π cm³ s⁻¹.

代入 r = 5，得到 dV/dt = 8π × 25 = 200π cm³ s⁻¹。

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10. Exam Strategy and Common Mistakes | 考试策略与常见错误

Success in Edexcel A-Level differentiation questions depends on accurate rule selection, clear working, and careful simplification.

在 Edexcel A-Level 微分题中取得好成绩，取决于正确选择法则、书写清晰以及仔细化简。

Rule | 法则	Use when | 使用场景
Chain rule | 链式法则	Function inside a function | 函数中有函数
Product rule | 乘法法则	Two functions multiplied | 两个函数相乘
Quotient rule | 除法法则	One function divided by another | 一个函数除以另一个函数

Common mistakes include forgetting to multiply by the inner derivative in the chain rule, reversing the terms in the quotient rule numerator, and failing to simplify negative or fractional powers before differentiating.

常见错误包括：在链式法则中忘记乘以内层导数，在除法法则分子中颠倒顺序，以及求导前没有先化简负指数或分数指数。

Always check the domain of the original function and the validity of stationary points. Present final answers in the form requested by the question.

始终检查原函数的定义域和驻点的有效性。按照题目要求的形式呈现最终答案。

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