Mastering Differentiation: Rules and Applications | 掌握微分：法则与应用

📚 Mastering Differentiation: Rules and Applications | 掌握微分：法则与应用

Differentiation is one of the core pillars of Edexcel A Level Pure Mathematics. It measures how a function changes as its input changes, and it is essential for problems involving gradients, rates of change, curve sketching and optimisation. This revision guide covers the key differentiation rules, standard results and common applications you need to master for the Edexcel exam.

微分是爱德思A Level纯数学的核心支柱之一。它衡量函数随自变量变化而变化的快慢，对于梯度、变化率、曲线作图与优化问题都至关重要。本复习指南涵盖爱德思考试中你必须掌握的主要微分法则、标准结论和常见应用。

1. First Principles and the Definition of the Derivative | 导数定义与第一原理

The derivative of a function gives the slope of the tangent to the curve at any point. From first principles, the derivative of f(x) is defined by the limit of the average rate of change as h tends to zero.

函数的导数给出曲线上任意一点处切线的斜率。根据第一原理，f(x) 的导数定义为当 h 趋向于零时平均变化率的极限。

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

For example, if f(x) = x², expanding f(x + h) = (x + h)² gives x² + 2xh + h². Subtracting x² and dividing by h leaves 2x + h, and taking the limit as h → 0 gives 2x.

例如，若 f(x) = x²，展开 f(x + h) = (x + h)² 得到 x² + 2xh + h²。减去 x² 再除以 h 得到 2x + h，令 h → 0 取极限，结果为 2x。

2. The Power Rule and Basic Polynomial Differentiation | 幂法则与基本多项式微分

The power rule is the most frequently used differentiation rule. For any real constant n, the derivative of xⁿ is n multiplied by x raised to the power n − 1.

幂法则是最常用的微分法则。对于任意实数常数 n，xⁿ 的导数为 n 乘以 x 的 n − 1 次幂。

d/dx (xⁿ) = n xⁿ⁻¹

This rule works for positive powers, negative powers and fractional powers. For example, d/dx (x⁵) = 5x⁴, while d/dx (1/x²) = d/dx (x⁻²) = −2x⁻³. You should also be able to differentiate roots by rewriting them as fractional powers, such as √x = x^½, so d/dx (√x) = ½ x⁻^½.

该法则适用于正整数幂、负整数幂和分数幂。例如 d/dx (x⁵) = 5x⁴，而 d/dx (1/x²) = d/dx (x⁻²) = −2x⁻³。你还需要将根式改写为分数幂来求导，例如 √x = x^½，所以 d/dx (√x) = ½ x⁻^½。

3. The Chain Rule for Composite Functions | 复合函数的链式法则

Many Edexcel exam functions are composite, meaning one function is applied inside another. The chain rule differentiates the outer function first, then multiplies by the derivative of the inner function.

爱德思考试中的许多函数都是复合函数，即一个函数嵌套在另一个函数内部。链式法则先对外层函数求导，再乘以内层函数的导数。

dy/dx = dy/du × du/dx

For example, to differentiate y = (2x + 1)³, set u = 2x + 1. Then y = u³, so dy/du = 3u² and du/dx = 2. Therefore dy/dx = 3(2x + 1)² × 2 = 6(2x + 1)².

例如，要对 y = (2x + 1)³ 求导，设 u = 2x + 1。则 y = u³，因此 dy/du = 3u²，du/dx = 2。所以 dy/dx = 3(2x + 1)² × 2 = 6(2x + 1)²。

4. The Product Rule | 乘积法则

When a function is written as the product of two simpler functions, you cannot simply differentiate each part separately. The product rule is used when y = u(x) × v(x).

当一个函数写成两个较简单函数的乘积时，不能简单地分别对各部分求导。当 y = u(x) × v(x) 时，应使用乘积法则。

dy/dx = u × dv/dx + v × du/dx

A typical example is 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. In Edexcel papers, product rule questions often also require simplification by taking out common factors.

一个典型例子是 y = x² sin x。设 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，所以 dy/dx = x² cos x + 2x sin x。在爱德思考卷中，乘积法则题目通常还需要通过提取公因式进行化简。

5. The Quotient Rule | 商法则

If y is a fraction with a function in the numerator and a function in the denominator, the quotient rule is required. For y = u / v, the derivative keeps the denominator squared and subtracts the product of u and the derivative of v.

如果 y 是一个分子和分母均为函数的分式，则需要使用商法则。对于 y = u / v，导数保持分母平方，并减去 u 与 v 的导数的乘积。

dy/dx = (v × du/dx − u × dv/dx) / v²

For example, if y = (x² + 1) / (x − 1), set u = x² + 1 and v = x − 1. Then du/dx = 2x, dv/dx = 1, so dy/dx = [ (x − 1)(2x) − (x² + 1)(1) ] / (x − 1)² = (2x² − 2x − x² − 1) / (x − 1)² = (x² − 2x − 1) / (x − 1)².

例如，若 y = (x² + 1) / (x − 1)，设 u = x² + 1，v = x − 1。则 du/dx = 2x，dv/dx = 1，所以 dy/dx = [ (x − 1)(2x) − (x² + 1)(1) ] / (x − 1)² = (2x² − 2x − x² − 1) / (x − 1)² = (x² − 2x − 1) / (x − 1)²。

6. Differentiating Exponentials and Logarithms | 指数函数与对数函数的微分

Exponential and logarithmic functions appear regularly in Edexcel Pure Mathematics. The natural exponential function eˣ is unique because its derivative is itself, while the derivative of ln x is the reciprocal of x.

指数函数和对数函数在爱德思纯数学中经常出现。自然指数函数 eˣ 的特殊性在于其导数等于它本身，而 ln x 的导数为 x 的倒数。

d/dx (eᵏˣ) = k eᵏˣ

d/dx (ln x) = 1/x

For a composite logarithmic function, use the chain rule. For example, d/dx [ ln(3x² + 1) ] = 6x / (3x² + 1). This is because the derivative of the inside 3x² + 1 is 6x, divided by the original inside function.

对于复合对数函数，需使用链式法则。例如 d/dx [ ln(3x² + 1) ] = 6x / (3x² + 1)。这是因为内层函数 3x² + 1 的导数为 6x，再除以原来的内层函数。

7. Differentiating Trigonometric Functions | 三角函数的微分

The standard derivatives of sine, cosine and tangent must be memorised for Edexcel A Level. They are especially important when combined with the chain rule, product rule or quotient rule.

爱德思A Level要求牢记正弦、余弦和正切的标准导数。它们在结合链式法则、乘积法则或商法则时尤为重要。

d/dx (sin x) = cos x

d/dx (cos x) = −sin x

d/dx (tan x) = sec² x

When the angle is not simply x, multiply by the derivative of the angle. For example, d/dx [ sin(2x) ] = 2 cos(2x), and d/dx [ cos(3x − 1) ] = −3 sin(3x − 1).

当角度不是简单的 x 时，需要乘上角度的导数。例如 d/dx [ sin(2x) ] = 2 cos(2x)，d/dx [ cos(3x − 1) ] = −3 sin(3x − 1)。

8. Second Derivatives and Rates of Change | 二阶导数与变化率

The second derivative is obtained by differentiating the first derivative. It tells you how quickly the gradient itself is changing and is used to classify stationary points and interpret acceleration.

二阶导数由一阶导数再次求导得到。它反映梯度本身变化的快慢，用于判断驻点类型以及解释加速度等实际变化率。

f”(x) = d²y/dx² = d/dx (dy/dx)

For example, if y = x³ − 3x² + 2x, then dy/dx = 3x² − 6x + 2 and d²y/dx² = 6x − 6. In kinematics, if displacement s = t³ − 2t² + 5t, then velocity is ds/dt = 3t² − 4t + 5 and acceleration is d²s/dt² = 6t − 4.

例如，若 y = x³ − 3x² + 2x，则 dy/dx = 3x² − 6x + 2，d²y/dx² = 6x − 6。在运动学中，若位移 s = t³ − 2t² + 5t，则速度为 ds/dt = 3t² − 4t + 5，加速度为 d²s/dt² = 6t − 4。

9. Stationary Points and Curve Sketching | 驻点与曲线作图

Stationary points occur where the first derivative is zero. Setting dy/dx = 0 and solving gives the x-coordinates of maximum points, minimum points or points of inflection.

驻点出现在一阶导数为零的位置。令 dy/dx = 0 并求解，可得到极大值点、极小值点或拐点的 x 坐标。

dy/dx = 0 at a stationary point

To classify a stationary point, substitute its x-coordinate into the second derivative. If d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum. If the second derivative equals zero, use a sign table to test the gradient on either side.

要判断驻点类型，可将其 x 坐标代入二阶导数。若 d²y/dx² > 0，该点为局部极小值；若 d²y/dx² < 0，则为局部极大值。若二阶导数等于零，则使用符号表检验驻点两侧的梯度变化。

10. Common Exam Mistakes and Strategy | 常见考试错误与答题策略

Edexcel differentiation questions often hide simple mistakes. Many students forget to multiply by the derivative of the inner function, use the quotient rule with the wrong sign, or fail to simplify before differentiating.

爱德思微分题目中常隐藏着简单错误。许多学生忘记乘以内层函数的导数，使用商法则时符号出错，或求导前没有先化简表达式。

Always rewrite roots and fractions as powers before differentiating, such as 1/x³ = x⁻³ and √x = x^½.

求导前务必先将根式和分式改写为幂的形式，例如 1/x³ = x⁻³，√x = x^½。
Apply the chain rule whenever the function is composite, especially with sin, cos, ln and eˣ.

只要函数是复合函数，就要使用链式法则，尤其是 sin、cos、ln 和 eˣ 的复合形式。
Check quotient rule signs carefully: the numerator is v du/dx minus u dv/dx, not the other way round.

仔细检查商法则的符号：分子是 v du/dx 减去 u dv/dx，而不是颠倒顺序。
Show full working so method marks can be awarded even if a numerical slip occurs.

展示完整的解题过程，这样即使出现数值计算错误，也能获得方法分。

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Mastering Differentiation: Rules and Applications | 掌握微分：法则与应用