A-Level Edexcel Maths: Mastering Differentiation | A-Level Edexcel 数学：掌握微分

📚 A-Level Edexcel Maths: Mastering Differentiation | A-Level Edexcel 数学：掌握微分

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics, appearing across Pure Mathematics papers and forming the backbone of applications in kinematics, optimisation and curve sketching.

微分是 Edexcel A-Level 数学中考查最频繁的主题之一，贯穿纯数学各卷，也是运动学、优化问题和曲线作图应用的基础。

1. The Derivative from First Principles | 从第一性原理求导

The derivative measures the instantaneous rate of change of a function. Edexcel regularly asks candidates to prove simple derivatives from first principles by considering the limiting value of a chord gradient.

导数衡量函数的瞬时变化率。Edexcel 经常要求考生通过考虑弦梯度的极限值，从第一性原理证明简单函数的导数。

f'(x) = lim (h → 0) [f(x + h) − f(x)] / h

To differentiate f(x) = x², expand (x + h)², simplify, divide by h, and then let h tend to 0. The result should always be f'(x) = 2x.

对 f(x) = x² 求导时，先展开 (x + h)²，化简后除以 h，再令 h 趋于 0，结果应为 f'(x) = 2x。

Examiners award marks for correct limit notation, correct expansion and explicit evaluation of the limit, so do not skip the line where h → 0.

考官会给正确的极限符号、正确的展开式以及明确写出 h → 0 的那一步赋分，因此不要省略代入极限的过程。

2. Power Rule and Basic Derivatives | 幂函数法则与基本导数

For any real power n, the derivative of xⁿ is n xⁿ⁻¹. This rule is the foundation for most polynomial differentiation questions in Edexcel papers.

对任意实数次幂 n，xⁿ 的导数为 n xⁿ⁻¹。这一法则是 Edexcel 试卷中大多数多项式求导题的基础。

d/dx (xⁿ) = n xⁿ⁻¹

You must also memorise the derivatives of standard functions: eˣ stays eˣ, ln x becomes 1/x, sin x becomes cos x, cos x becomes −sin x, and aˣ becomes aˣ ln a.

你还必须熟记标准函数的导数：eˣ 的导数仍为 eˣ，ln x 的导数为 1/x，sin x 的导数为 cos x，cos x 的导数为 −sin x，aˣ 的导数为 aˣ ln a。

Always rewrite square roots, reciprocal powers and rational terms as powers before differentiating. For example, √x becomes x^½ and 1/x³ becomes x⁻³.

求导前始终要把平方根、倒数幂和有理项改写为幂的形式。例如 √x 可写成 x^½，1/x³ 可写成 x⁻³。

3. Chain Rule | 链式法则

The chain rule is used when one function is applied inside another. Let y be a function of u, and u be a function of x, so that dy/dx = dy/du × du/dx.

当一个函数嵌套在另一个函数内部时，就要使用链式法则。设 y 是 u 的函数，u 是 x 的函数，则 dy/dx = dy/du × du/dx。

dy/dx = dy/du × du/dx

For y = (3x² + 5)⁴, take u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 24x(3x² + 5)³.

对于 y = (3x² + 5)⁴，令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x，最终 dy/dx = 24x(3x² + 5)³。

Edexcel candidates often lose marks by forgetting to multiply by the derivative of the inner function, so always check for a chain rule when the function is composite.

Edexcel 考生常常因为忘记乘以内层函数的导数而丢分，因此遇到复合函数时一定要检查是否需要使用链式法则。

4. Product Rule | 乘积法则

When y is the product of two functions, y = uv, the derivative is given by the product rule. This is essential for functions such as x² sin x or eˣ cos x.

当 y 是两个函数的乘积，即 y = uv 时，其导数由乘积法则给出。这对于 x² sin x 或 eˣ cos x 等函数至关重要。

dy/dx = u dv/dx + v du/dx

For y = x² sin x, let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x, so dy/dx = x² cos x + 2x sin x.

对于 y = x² sin x，令 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，因此 dy/dx = x² cos x + 2x sin x。

It is good practice to state u, v, du/dx and dv/dx clearly before substituting, because this structured method reduces sign and factor errors.

最好先明确写出 u、v、du/dx 和 dv/dx，再代入公式，这种有条理的方法可以减少符号和系数错误。

5. Quotient Rule | 商法则

If y = u/v, where u and v are functions of x, then the derivative is found by subtracting v du/dx from u dv/dx and dividing by v².

如果 y = u/v，其中 u 和 v 都是 x 的函数，那么导数等于 u dv/dx 减去 v du/dx，再除以 v²。

dy/dx = (v du/dx − u dv/dx) / v²

For y = x / (x² + 1), take u = x and v = x² + 1. Then du/dx = 1 and dv/dx = 2x, giving dy/dx = (x² + 1 − 2x²) / (x² + 1)² = (1 − x²) / (x² + 1)².

对于 y = x / (x² + 1)，令 u = x，v = x² + 1。则 du/dx = 1，dv/dx = 2x，得到 dy/dx = (x² + 1 − 2x²) / (x² + 1)² = (1 − x²) / (x² + 1)²。

The most common examination error is writing u dv/dx − v du/dx in the numerator instead of v du/dx − u dv/dx, so memorise the order carefully.

最常见的考试错误是把分子写成 u dv/dx − v du/dx，而正确顺序应为 v du/dx − u dv/dx，因此要仔细牢记顺序。

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

Stationary points occur where dy/dx = 0. These can be local maxima, local minima or points of inflection, and Edexcel often asks for their coordinates and nature.

驻点出现在 dy/dx = 0 的位置。这些点可能是局部极大值、局部极小值或拐点，Edexcel 经常要求求出其坐标并判断性质。

f”(x) = d²y/dx²

To classify a stationary point, find the second derivative. If f”(a) > 0 the point is a minimum; if f”(a) < 0 it is a maximum; if f''(a) = 0, use the first derivative test.

要判断驻点性质，需要求二阶导数。若 f”(a) > 0，该点为极小值；若 f”(a) < 0，该点为极大值；若 f''(a) = 0，则应使用一阶导数检验法。

The first derivative test checks the sign of f'(x) on either side of the stationary point. A sign change from negative to positive indicates a minimum, and from positive to negative indicates a maximum.

一阶导数检验法检查驻点两侧 f'(x) 的符号。符号由负变正则表示极小值，由正变负则表示极大值。

7. Tangents and Normals | 切线与法线

The gradient of a tangent to a curve at a point is the value of dy/dx at that point. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal.

曲线在某点处切线的斜率等于该点处 dy/dx 的值。法线与切线垂直，因此其斜率为切线斜率的负倒数。

m_tangent = dy/dx, m_normal = −1 / m_tangent

To find the equation of a tangent or normal, use the point-slope form y − y₁ = m(x − x₁), where (x₁, y₁) is the point on the curve and m is the gradient.

求切线或法线方程时，使用点斜式 y − y₁ = m(x − x₁)，其中 (x₁, y₁) 是曲线上的点，m 是斜率。

Always substitute both x and y into the derivative to find the gradient; using just x or just the original equation can lead to incorrect answers.

务必同时将 x 和 y 代入导数中求斜率；只代入 x 或只使用原方程都可能导致答案错误。

8. Increasing and Decreasing Functions | 增函数与减函数

A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0. Edexcel questions often ask you to find the range of x for which a function is increasing.

在区间上若 f'(x) > 0，则函数递增；若 f'(x) < 0，则函数递减。Edexcel 题目经常要求找出函数递增时 x 的取值范围。

For example, if f'(x) = 3x² − 12, solve 3x² − 12 > 0 to obtain x < −2 or x > 2, so the function is increasing on those intervals.

例如，若 f'(x) = 3x² − 12，解 3x² − 12 > 0 得到 x < −2 或 x > 2，因此函数在这些区间上递增。

Remember that a function can be increasing on two separate intervals; always state them as a union or with ‘or’, not as a single combined inequality.

请记住，函数可能在两个不相连的区间上递增；要始终用并集或 ‘或’ 来表示，不要写成单一合并的不等式。

9. Optimisation Problems | 优化问题

Optimisation questions require you to create a formula for a quantity such as area, volume or cost, and then use differentiation to find its maximum or minimum value.

优化题要求你为面积、体积或成本等量建立公式，然后利用微分求出其最大值或最小值。

Express the quantity in terms of one variable using a constraint, differentiate, set the derivative equal to zero, and solve. Then use the second derivative to confirm the nature.

利用约束条件将待求量表示成单变量函数，求导并令导数为零，解方程。然后使用二阶导数确认其性质。

For a rectangular enclosure with fixed perimeter P, express area A in terms of one side x, find A'(x) = 0, and show A”(x) < 0 to prove a maximum.

对于周长为固定值 P 的矩形围栏，用一边长 x 表示面积 A，令 A'(x) = 0，并证明 A”(x) < 0 以确认最大值。

Always check the physical domain of the variable, such as x > 0 or 0 < x < total length, and verify that your answer lies within it.

始终检查变量的实际定义域，例如 x > 0 或 0 < x < 总长度，并验证所得答案确实位于该区间内。

10. Connected Rates of Change | 相关变化率

Connected rates of change are modelled using the chain rule. If a quantity A depends on r, and r changes with time t, then dA/dt = dA/dr × dr/dt.

相关变化率可用链式法则建立模型。如果量 A 依赖于 r，而 r 随时间 t 变化，则 dA/dt = dA/dr × dr/dt。

dA/dt = dA/dr × dr/dt

Typical Edexcel examples include an expanding circular stain where area A = πr², or a spherical balloon where volume V = (4/3)πr³.

Edexcel 典型例题包括扩散的圆形污渍，其面积 A = πr²，或球形气球，其体积 V = (4/3)πr³。

Identify which rate is given and which rate is required, then link them through an intermediate variable using the chain rule.

先确定已知的变化率和需要求的变化率，然后通过一个中间变量用链式法则将它们联系起来。

11. Parametric Differentiation | 参数微分

When a curve is defined parametrically by x = f(t) and y = g(t), the gradient dy/dx is found by dividing dy/dt by dx/dt.

当曲线由参数方程 x = f(t)、y = g(t) 定义时，斜率 dy/dx 等于 dy/dt 除以 dx/dt。

dy/dx = (dy/dt) / (dx/dt)

For example, if x = 3t² and y = 2t³, then dy/dx = 6t² / 6t = t. You can then substitute a given t value to find the gradient at a point.

例如，若 x = 3t² 且 y = 2t³，则 dy/dx = 6t² / 6t = t。然后可代入给定的 t 值求出某点的斜率。

For second derivatives in parametric form, use d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) × dt/dx, but this is only examined at the higher tier.

参数形式下的二阶导数为 d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) × dt/dx，但这仅在较高层级考查。

12. Common Exam Mistakes and Tips | 常见考试错误与技巧

One of the most frequent mistakes is applying the product rule when the expression is actually a simple constant multiple, or vice versa. Always inspect the structure before choosing a rule.

最常见的错误之一是在表达式实际上是简单常数倍时却使用乘积法则，或者相反。选择法则之前一定要先观察结构。

Another common error is forgetting to differentiate the inside function when using the chain rule. Write ‘inside function’ and ‘derivative of inside function’ explicitly if needed.

另一个常见错误是在使用链式法则时忘记对内层函数求导。如有需要，可明确写出 ‘内层函数’ 及其导数。

In quotient rule problems, careful signs are essential. Many candidates lose marks through a single minus sign error, so check the numerator order v du/dx − u dv/dx every time.

在商法则问题中，符号必须格外小心。许多考生因一个负号错误而丢分，因此每次都要检查分子顺序 v du/dx − u dv/dx。

Finally, always answer the specific question: state coordinates, units, intervals or equations exactly as requested, and justify the nature of stationary points with a derivative test.

最后，要始终按要求作答：明确写出坐标、单位、区间或方程，并用导数检验法说明驻点性质。

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A-Level Edexcel Maths: Mastering Differentiation | A-Level Edexcel 数学：掌握微分