📚 Differentiation Techniques for Edexcel A-Level Pure Mathematics | 爱德思 A-Level 纯数学：微分技巧
Differentiation is one of the central pillars of A-Level Pure Mathematics, and Edexcel exam papers test it in both routine and applied contexts. This revision guide builds the key derivative rules from first principles, moves through chain, product and quotient rules, then extends into implicit and parametric differentiation. Each section pairs an English explanation with a Chinese translation so you can check terminology and method in both languages.
微分是 A-Level 纯数学的重要支柱之一，爱德思考试既考查基本运算，也考查实际应用。本复习指南从第一原理出发构建核心导数规则，逐步过渡到链式法则、乘积法则与商法则，再扩展到隐函数微分和参数方程微分。每一节都提供中英文对照解释，帮助你熟悉术语和解题方法。
1. First Principles and Notation | 第一原理与导数记号
Differentiation from first principles defines the derivative as the limit of the average rate of change of a function. If y = f(x), then the derivative f'(x) is the limit of the gradient of a chord as the interval h tends to zero.
从第一原理出发，导数被定义为函数平均变化率的极限。若 y = f(x)，则导数 f'(x) 是当区间 h 趋近于零时弦的斜率的极限。
f'(x) = limₕ→₀ [f(x+h) − f(x)] / h
In Edexcel papers you will normally meet this definition in a short proof or a limit calculation, especially for simple powers such as f(x) = x². The notation dy/dx is equivalent to f'(x) and is called Leibniz notation, which is particularly useful in the chain rule and implicit differentiation.
在爱德思考试中，你通常会在简短的证明或极限计算中遇到这个定义，尤其是对 f(x) = x² 这样的简单幂函数。记号 dy/dx 等价于 f'(x)，称为莱布尼茨记号，它在链式法则和隐函数微分中特别有用。
2. Power Rule and Polynomials | 幂函数法则与多项式
The power rule is the most frequently used differentiation rule at A-Level. For any real power n, the derivative of xⁿ is n xⁿ⁻¹. This rule works for positive powers, negative powers and fractional powers, so it also covers reciprocal and square-root functions.
幂函数法则是 A-Level 中最常用的微分规则。对于任意实数指数 n，xⁿ 的导数是 n xⁿ⁻¹。这个规则适用于正幂、负幂和分数幂，因此它也涵盖倒数函数和平方根函数。
d/dx (xⁿ) = n xⁿ⁻¹
For example, if y = 5x⁴ − 3x² + 7x − 2, then dy/dx = 20x³ − 6x + 7. The constant term −2 differentiates to zero because horizontal lines have zero gradient. The table below summarises common power-rule cases.
例如，若 y = 5x⁴ − 3x² + 7x − 2，则 dy/dx = 20x³ − 6x + 7。常数项 −2 的导数为零，因为水平线的斜率为零。下表总结了常见的幂函数法则情形。
| Function | Derivative |
|---|---|
| x³ | 3x² |
| √x = x^½ | ½ x^−½ = 1 / (2√x) |
| 1/x = x^−1 | −x^−2 = −1 / x² |
Always write roots and denominators using indices before differentiating. This avoids mistakes when applying the power rule to expressions such as 1/x³ or √(x⁵).
在微分之前，一定要先把根式和分母写成指数形式。这样在将幂函数法则应用于 1/x³ 或 √(x⁵) 等表达式时，可以避免错误。
3. Chain Rule | 链式法则
The chain rule is used to differentiate composite functions, where one function is inside another. If y is a function of u, and u is a function of x, then the rate of change of y with respect to x is the product of the two rates of change.
链式法则用于对复合函数求导，即一个函数嵌套在另一个函数内部。如果 y 是 u 的函数，u 是 x 的函数，那么 y 对 x 的变化率等于这两个变化率的乘积。
dy/dx = dy/du × du/dx
A typical Edexcel example is y = (3x² + 1)⁵. Let u = 3x² + 1, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x, giving dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴.
爱德思考试中的典型例子是 y = (3x² + 1)⁵。设 u = 3x² + 1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴。
When a function is linear, such as y = (ax + b)ⁿ, the chain rule gives the quick result dy/dx = an(ax + b)ⁿ⁻¹. This shortcut is used regularly in integration and in solving differential equations.
当内层函数是线性的，例如 y = (ax + b)ⁿ，链式法则可直接给出 dy/dx = an(ax + b)ⁿ⁻¹。这个快捷结果在积分和微分方程中经常使用。
4. Product Rule | 乘积法则
The product rule is used when two different functions of x are multiplied together. It states that the derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.
当两个不同的 x 函数相乘时，需要使用乘积法则。它的含义是：乘积的导数等于第一个函数的导数乘以第二个函数，再加上第一个函数乘以第二个函数的导数。
d/dx (uv) = u’v + uv’
For example, if y = x² eˣ, let u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, so dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x).
例如，若 y = x² eˣ，设 u = x²，v = eˣ。则 u’ = 2x，v’ = eˣ，因此 dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x)。
The product rule can be extended to three functions, but at A-Level it is usually easier to group two factors at a time. Always factorise the final answer where possible, because exam mark schemes often reward the simplified form.
乘积法则可以扩展到三个函数的乘积，但在 A-Level 中通常每次将两个因子组合起来更简单。只要有可能，就应该对最终结果进行因式分解，因为考试评分方案通常会奖励简化形式。
5. Quotient Rule | 商法则
The quotient rule is used to differentiate a fraction where the numerator and denominator are both functions of x. It is often tested with trigonometric, exponential or logarithmic functions in the numerator and denominator.
商法则用于对分子和分母都是 x 的函数的分数求导。它经常在分子和分母包含三角函数、指数函数或对数函数时考查。
dy/dx = (v u’ − u v’) / v²
For y = (x² + 1)/(x − 2), let u = x² + 1 and v = x − 2. Then u’ = 2x, v’ = 1, and dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1) / (x − 2)² = (x² − 4x − 1) / (x − 2)².
对于 y = (x² + 1)/(x − 2)，设 u = x² + 1，v = x − 2。则 u’ = 2x，v’ = 1，所以 dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1) / (x − 2)² = (x² − 4x − 1) / (x − 2)²。
A common error is reversing the subtraction in the numerator. Remember: it is v u’ first, then minus u v’. Writing u and v clearly at the start of the question helps prevent sign mistakes.
一个常见错误是把分子中的减法顺序写反。请记住：先写 v u’，再减去 u v’。在解题开始时清楚地写出 u 和 v，有助于避免符号错误。
6. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The exponential function eˣ is unique because its derivative is itself. More generally, when the exponent is a linear function of x, the derivative introduces a constant factor.
指数函数 eˣ 的特殊之处在于它的导数就是它本身。更一般地，当指数是 x 的线性函数时，导数会引入一个常数因子。
d/dx (eˣ) = eˣ
d/dx (eᵃˣ) = a eᵃˣ
For natural logarithms, the derivative of ln x is 1/x for x > 0. Because ln(kx) = ln k + ln x, the derivative of ln(kx) is also 1/x, since ln k is a constant.
对于自然对数，当 x > 0 时，ln x 的导数是 1/x。由于 ln(kx) = ln k + ln x，因此 ln(kx) 的导数也是 1/x，因为 ln k 是常数。
d/dx (ln x) = 1/x
For example, if y = e³ˣ + ln(2x), then dy/dx = 3e³ˣ + 1/x. Watch for these functions inside product, quotient or chain rule questions, where the exponential term often remains unchanged during differentiation.
例如，若 y = e³ˣ + ln(2x)，则 dy/dx = 3e³ˣ + 1/x。在乘积法则、商法则或链式法则题目中要注意这些函数，因为指数函数求导后通常保持不变。
7. Differentiating Trigonometric Functions | 三角函数的微分
The three core trigonometric derivatives must be memorised. For all A-Level pure mathematics papers, these results are assumed knowledge and appear in many different question types.
三个核心三角函数的导数必须牢记。在 A-Level 纯数学考试中，这些结果属于必备知识，会出现在许多不同类型的题目中。
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
The reciprocal trigonometric functions also have standard derivatives. These can be derived using the quotient rule, but it is faster to know them directly for Edexcel papers.
倒数三角函数也有标准导数。它们可以用商法则推导，但在爱德思考试中直接记住会更快。
| Function | Derivative |
|---|---|
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| cot x | −cosec² x |
When the angle is not simply x but ax + b, apply the chain rule. For instance, d/dx [sin(2x)] = 2 cos(2x), and d/dx [cos(5x − 3)] = −5 sin(5x − 3).
当角度不是简单的 x，而是 ax + b 时，需要应用链式法则。例如 d/dx [sin(2x)] = 2 cos(2x)，d/dx [cos(5x − 3)] = −5 sin(5x − 3)。
8. Implicit Differentiation | 隐函数微分
Implicit differentiation is used when y is not written explicitly as a function of x. Instead of solving for y first, differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule to any y term.
当 y 没有被明确写成 x 的函数时，就需要使用隐函数微分。不要先解出 y，而是对等式两边关于 x 求导，将 y 视为 x 的函数，并对每个含 y 的项应用链式法则。
d/dx (y²) = 2y dy/dx
For example, for the circle x² + y² = 25, differentiating both sides gives 2x + 2y dy/dx = 0. Rearranging gives dy/dx = −x/y.
例如，对于圆 x² + y² = 25，两边求导得到 2x + 2y dy/dx = 0。整理后得到 dy/dx = −x/y。
Implicit differentiation is especially useful for equations involving products of x and y, or x and y terms mixed together. In Edexcel exam questions, you are often asked to find dy/dx and then evaluate it at a given point.
隐函数微分对于含有 x 与 y 的乘积或混合项的方程特别有用。在爱德思考试题中，通常要求先求出 dy/dx，然后在给定点处计算其值。
9. Parametric Differentiation | 参数方程微分
When x and y are both given in terms of a third variable t, the curve is defined parametrically. The gradient dy/dx is found by dividing the derivative of y with respect to t by the derivative of x with respect to t.
当 x 和 y 都用第三个变量 t 表示时，曲线通过参数方程定义。斜率 dy/dx 可以通过 y 对 t 的导数除以 x 对 t 的导数来求得。
dy/dx = (dy/dt) ÷ (dx/dt)
For example, if x = t² + 1 and y = t³ − 2t, then dx/dt = 2t and dy/dt = 3t² − 2. Therefore dy/dx = (3t² − 2) / (2t).
例如，若 x = t² + 1，y = t³ − 2t，则 dx/dt = 2t，dy/dt = 3t² − 2。因此 dy/dx = (3t² − 2) / (2t)。
The second derivative for parametric equations is not found by simply differentiating dy/dx with respect to t. You must use d²y/dx² = d/dt(dy/dx) ÷ (dx/dt).
参数方程的二阶导数不能仅仅对 dy/dx 关于 t 直接求导。必须使用 d²y/dx² = d/dt(dy/dx) ÷ (dx/dt)。
10. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative measures the rate of change of the first derivative. It is found by differentiating dy/dx again with respect to x. For y = x³ − 3x² + 2, the first derivative is dy/dx = 3x² − 6x, and the second derivative is d²y/dx² = 6x − 6.
二阶导数衡量一阶导数的变化率。它是将 dy/dx 再对 x 求导得到的。对于 y = x³ − 3x² + 2，一阶导数为 dy/dx = 3x² − 6x，二阶导数为 d²y/dx² = 6x − 6。
The sign of d²y/dx² tells you about the curvature of a graph. If d²y/dx² > 0, the curve is concave upwards, and if d²y/dx² < 0, the curve is concave downwards. This idea is linked to classifying stationary points.
d²y/dx² 的符号表示图形的弯曲方向。若 d²y/dx² > 0，曲线向上凹；若 d²y/dx² < 0，曲线向下凹。这一概念与驻点的分类密切相关。
11. Tangents, Normals and Stationary Points | 切线、法线与驻点
To find the equation of a tangent to a curve at a point, first substitute the x-coordinate into the first derivative to get the gradient m. Then use y − y₁ = m(x − x₁) with the given point.
求曲线在某点处的切线方程时，首先将该点的 x 坐标代入一阶导数得到斜率 m。然后利用 y − y₁ = m(x − x₁) 和给定的点写出方程。
The normal to a curve is perpendicular to the tangent. Its gradient is the negative reciprocal of the tangent gradient: m_normal = −1/m_tangent. If the tangent has gradient zero, the normal is a vertical line.
曲线的法线与切线垂直。法线的斜率是切线斜率的负倒数：m_normal = −1/m_tangent。如果切线斜率为零，则法线是一条竖直直线。
Stationary points occur where dy/dx = 0. There are three main types: local maximum, local minimum, and point of inflection. The second derivative test classifies a stationary point as a maximum if d²y/dx² < 0, or a minimum if d²y/dx² > 0.
驻点出现在 dy/dx = 0 处。主要有三种类型：局部极大值、局部极小值和拐点。二阶导数判别法可对驻点进行分类：若 d²y/dx² < 0 为极大值，若 d²y/dx² > 0 为极小值。
12. Exam-Style Problem Solving | 考试题型策略
In Edexcel A-Level maths exams, differentiation questions often combine several rules in one problem. Start by identifying the overall structure: is the expression a product, a quotient, or a composite function? Then choose the appropriate rule before writing any derivatives.
在爱德思 A-Level 数学考试中，微分题往往在一道题中综合考查多个规则。首先要识别整体结构：这个表达式是乘积、商还是复合函数？然后在写出任何导数之前选择合适的规则。
For structured questions, define u and v clearly when using the product or quotient rule. For implicit questions, remember to multiply every y derivative by dy/dx. For parametric questions, do not forget to divide by dx/dt.
对于结构化题目，在使用乘积法则或商法则时要清楚地定义 u 和 v。对于隐函数题，要记住每个 y 的导数都要乘以 dy/dx。对于参数方程题，不要忘记除以 dx/dt。
Finally, always simplify your derivative where possible, and check that your answer is dimensionally and algebraically sensible. Substituting a small test value can
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