Differentiation: Techniques and Applications | 微分：技巧与应用

📚 Differentiation: Techniques and Applications | 微分：技巧与应用

Differentiation is at the heart of A-Level Edexcel Pure Mathematics. It gives a precise way to measure how a function changes at an instant, leading directly to tangents, stationary points, optimisation and connected rates. This revision guide covers the core techniques and the most common exam applications.

微分是 A-Level Edexcel 纯数学的核心内容。它提供了一种精确测量函数在某一瞬间变化的方法，并由此引出切线、驻点、最优化以及相关变化率。本复习指南涵盖核心技巧和最常见的考试应用。

1. First Principles and Key Notation | 第一性原理与核心记号

The derivative of a function f(x) at a point is defined as the limit of the average rate of change as the interval tends to zero.

函数 f(x) 在某一点的导数定义为当区间趋于零时平均变化率的极限。

f'(x) = lim h→0 [f(x + h) − f(x)] / h

In Edexcel questions you may be asked to use this definition to differentiate simple functions such as x² or x³. This is called differentiation from first principles.

在 Edexcel 考题中，你可能需要用此定义对 x² 或 x³ 等简单函数求导。这称为第一性原理微分。

The notation dy/dx is read as ‘the rate of change of y with respect to x’. It is not a fraction in elementary work, but it behaves usefully in the chain rule and related rates.

记号 dy/dx 读作“y 对 x 的变化率”。在基础内容中它不是分数，但在链式法则和相关变化率中它表现出类似分数的实用性质。

2. Basic Rules: Power, Constant Multiple and Sum | 基本法则：幂函数、常数倍、和差

For any real constant n, the power rule states that the derivative of xⁿ is n xⁿ⁻¹. This is the most frequently used rule in A-Level differentiation.

对于任意实数常数 n，幂函数法则指出 xⁿ 的导数为 n xⁿ⁻¹。这是 A-Level 微分中最常用的法则。

d/dx (xⁿ) = n xⁿ⁻¹

The constant multiple rule says that if y = a f(x), where a is a constant, then dy/dx = a f'(x). The sum rule allows term-by-term differentiation.

常数倍法则指出，如果 y = a f(x)，其中 a 为常数，则 dy/dx = a f'(x)。和差法则允许逐项求导。

If y = 4x⁵, then dy/dx = 20x⁴.
如果 y = 4x⁵，则 dy/dx = 20x⁴。
If y = 3x² − 2x + 7, then dy/dx = 6x − 2.
如果 y = 3x² − 2x + 7，则 dy/dx = 6x − 2。

Always rewrite rational or root expressions in index form before differentiating. For example, 1/x² becomes x⁻² and √x becomes x^½.

求导前始终把有理式或根式改写成指数形式。例如，1/x² 写作 x⁻²，√x 写作 x^½。

3. Product and Quotient Rules | 积法则与商法则

When two functions of x are multiplied, the product rule is used. If y = u v, where u and v are functions of x, then the derivative is given by:

当两个关于 x 的函数相乘时，需使用积法则。如果 y = u v，其中 u 和 v 是关于 x 的函数，则导数为：

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

The quotient rule applies when one function is divided by another. If y = u / v, then:

当一个函数除以另一个函数时，使用商法则。如果 y = u / v，则：

dy/dx = (v du/dx − u dv/dx) / v²

In exam work, it is useful to state u, v, du/dx and dv/dx clearly before substituting into the formula. This reduces sign errors and makes method marks easier to obtain.

在考试中，建议先明确写出 u、v、du/dx 和 dv/dx，再代入公式。这可以减少符号错误，也更容易获得方法分。

4. Chain Rule and Rate of Change | 链式法则与变化率

The chain rule is used for composite functions. If y is a function of u and u is a function of x, then:

链式法则用于复合函数。如果 y 是 u 的函数，u 是 x 的函数，则：

dy/dx = dy/du × du/dx

This rule allows you to differentiate expressions such as (3x² + 5)⁴, sin(2x) or ln(x² + 1). Differentiate the outer function first, then multiply by the derivative of the inner function.

该法则使你可以对 (3x² + 5)⁴、sin(2x) 或 ln(x² + 1) 等表达式求导。先对外层函数求导，再乘以内层函数的导数。

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

例如，如果 y = (3x² + 5)⁴，令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x，故 dy/dx = 4(3x² + 5)³ × 6x。

5. Differentiating 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. This property is central to growth and decay problems in the Edexcel specification.

指数函数 eˣ 具有独特性质：其导数等于它本身。这一性质是 Edexcel 考试范围内增长与衰减问题的核心。

d/dx (eˣ) = eˣ

For a constant k, the chain rule gives d/dx (e^k x) = k e^k x. The natural logarithm function ln x has derivative 1/x for x > 0.

对于常数 k，由链式法则可得 d/dx (e^k x) = k e^k x。自然对数函数 ln x 在 x > 0 时的导数为 1/x。

d/dx (ln x) = 1/x

More generally, the derivative of ln(f(x)) is f'(x) / f(x). This result appears frequently when differentiating logarithmic functions.

更一般地，ln(f(x)) 的导数为 f'(x) / f(x)。该结果在求对数函数的导数时经常出现。

6. Trigonometric Differentiation | 三角函数的微分

The derivatives of the three basic trigonometric functions must be memorised. They are used directly and within the chain rule, product rule and quotient rule.

三个基本三角函数的导数必须牢记。它们可以直接使用，也可以与链式法则、积法则和商法则一起使用。

d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x

For a linear inner function ax + b, the chain rule introduces a factor of a. For example, d/dx(sin 2x) = 2 cos 2x and d/dx(cos 3x) = −3 sin 3x.

对于线性内层函数 ax + b，链式法则会引入因子 a。例如，d/dx(sin 2x) = 2 cos 2x，d/dx(cos 3x) = −3 sin 3x。

In Edexcel papers, trigonometric differentiation often appears in contexts such as modelling, tangent equations and optimisation. Accuracy with signs is essential.

在 Edexcel 试卷中，三角函数的微分常出现在建模、切线方程和最优化等情境中。正确处理符号至关重要。

7. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative is obtained by differentiating the first derivative. It is written as d²y/dx² or f”(x). It measures the rate of change of the gradient.

二阶导数由一阶导数再次求导得到，记作 d²y/dx² 或 f”(x)。它衡量梯度的变化率。

d²y/dx² = d/dx (dy/dx)

If d²y/dx² > 0 at a point, the curve is concave up, and a stationary point there is a local minimum. If d²y/dx² < 0, the curve is concave down, and a stationary point there is a local maximum.

如果某点处 d²y/dx² > 0，曲线在该点凹向上，该点处的驻点为局部极小值。如果 d²y/dx² < 0，曲线凹向下，该点处的驻点为局部极大值。

When the second derivative equals zero, you should use the first derivative test by checking the sign of dy/dx on either side of the stationary point.

当二阶导数等于零时，应使用一阶导数判别法，检查驻点两侧 dy/dx 的符号。

8. Tangents and Normals | 切线与法线

At a point where x = a, the gradient of the tangent to the curve y = f(x) is given by f'(a). The tangent is the straight line passing through the point with this gradient.

在 x = a 处，曲线 y = f(x) 的切线梯度为 f'(a)。切线是经过该点且具有该梯度的直线。

y − y₁ = m(x − x₁)

Here (x₁, y₁) is the point on the curve and m = f'(x₁) is the gradient of the tangent.

这里 (x₁, y₁) 是曲线上的点，m = f'(x₁) 是切线的梯度。

The normal is perpendicular to the tangent. Its gradient is the negative reciprocal of the tangent gradient:

法线垂直于切线。其梯度是切线梯度的负倒数：

m_normal = −1 / m_tangent

Always calculate the y-coordinate as well as the gradient before writing the equation of a tangent or normal. A common error is using only the x-coordinate.

在写出切线或法线方程之前，务必同时计算 y 坐标和梯度。常见错误是只使用 x 坐标。

9. Stationary Points and Classification | 驻点与分类

Stationary points occur where the first derivative is zero. Solving dy/dx = 0 gives the x-coordinates of these points.

驻点出现在一阶导数为零的位置。解方程 dy/dx = 0 可得到这些点的 x 坐标。

dy/dx = 0

To classify a stationary point, substitute the x-value into the second derivative. A positive result indicates a minimum, and a negative result indicates a maximum.

要判断驻点类型，将 x 值代入二阶导数。结果为正表示极小值点，结果为负表示极大值点。

Alternatively, examine the sign of dy/dx just to the left and right of the stationary point. If the gradient changes from positive to negative, the point is a maximum; if it changes from negative to positive, the point is a minimum.

也可以检查驻点左侧和右侧 dy/dx 的符号。如果梯度由正变负，该点为极大值点；如果由负变正，该点为极小值点。

Remember to find the full coordinates by substituting the x-value back into the original equation y = f(x).

记住将 x 值代回原方程 y = f(x) 以求出完整坐标。

10. Optimisation Problems | 最优化问题

Optimisation questions ask you to maximise or minimise a quantity such as area, volume, cost or distance. The key is to express this quantity as a function of one variable.

最优化问题要求使面积、体积、成本或距离等量最大化或最小化。关键是把这个量表示为单个变量的函数。

Then differentiate the function, set the derivative equal to zero and solve for the variable. Use the second derivative or sign test to confirm that the solution is a maximum or minimum.

然后对函数求导，令导数为零并解出变量。使用二阶导数或符号判别法确认解是最大值还是最小值。

Always check the domain of the problem. For example, lengths and volumes cannot be negative, so an answer outside the valid domain must be rejected.

始终检查问题的定义域。例如，长度和体积不能为负，因此超出有效定义域的答案必须舍去。

Edexcel questions often require you to use a constraint equation to eliminate one variable before differentiation. Write the constraint clearly to avoid algebraic mistakes.

Edexcel 题目通常要求先利用约束方程消去一个变量，再求导。清晰写出约束条件以避免代数错误。

11. Connected Rates of Change | 相关变化率

Related rates problems link two or more rates using the chain rule. For example, a volume V may depend on radius r, and radius may depend on time t.

相关变化率问题利用链式法则将两个或多个变化率联系起来。例如，体积 V 可能依赖于半径 r，而半径又依赖于时间 t。

dV/dt = dV/dr × dr/dt

The procedure is to identify the known rate, find the missing derivative from a formula, and then multiply the two derivatives together.

解题步骤是确定已知变化率，从公式中求出缺失的导数，然后将两个导数相乘。

Pay attention to signs: increasing quantities give positive rates, while decreasing quantities give negative rates. Units must also be consistent.

注意符号：增大的量给出正变化率，减小的量给出负变化率。单位也必须保持一致。

12. Common Exam Pitfalls | 常见考试失分点

Many differentiation errors come from missing minus signs, especially when differentiating cos x or when applying the quotient rule. Write each step in full and double-check signs.

许多微分错误来自漏掉负号，尤其是对 cos x 求导或使用商法则时。每一步都要完整写出并仔细检查符号。

Another common mistake is forgetting to multiply by the derivative of the inner function in chain rule problems. Always ask: what is the inner function, and what is its derivative?

另一个常见错误是在链式法则问题中忘记乘以内层函数的导数。始终要问：内层函数是什么？它的导数是什么？

Do not write dy/dx = cos 2x for y = sin 2x; the correct derivative is dy/dx = 2 cos 2x.
不要对 y = sin 2x 写出 dy/dx = cos 2x；正确导数为 dy/dx = 2 cos 2x。
Do not forget to substitute the y-coordinate when finding a tangent equation.
求切线方程时不要忘记代入 y 坐标。
Do not classify a stationary point without evaluating the second derivative or checking the sign change.
未计算二阶导数或检查符号变化，就不要对驻点进行分类。

Practice past-paper questions under timed conditions. Consistent method, clear working and careful notation are the best ways to secure full marks in Edexcel differentiation questions.

在定时条件下练习历年真题。一致的方法、清晰的步骤和规范的记号是确保在 Edexcel 微分题中获得满分的最佳途径。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Differentiation: Techniques and Applications | 微分：技巧与应用