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 forms the backbone of A-Level calculus, enabling us to analyse how functions change, find gradients, and solve real-world problems. This article systematically covers all the essential rules and applications required for the Edexcel specification, from basic principles to advanced parametric and implicit techniques.

微分是 A-Level 微积分的核心，帮助我们分析函数的变化方式、求梯度以及解决现实问题。本文系统覆盖 Edexcel 考纲要求的所有基础法则与应用，从基本法则到高级的参数方程与隐函数技巧，帮助你在考试中稳操胜券。

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1. The Idea of Differentiation | 微分的基本概念

lly, the derivative dy/dx represents the gradient of the tangent to the curve at any point. In physics, if y is displacement, dy/dx gives velocity.

微分衡量函数 y = f(x) 关于 x 的瞬时变化率。从几何角度看，导数 dy/dx 表示曲线上任意一点切线的梯度。在物理中，如果 y 代表位移，则 dy/dx 给出速度。

he process of finding dy/dx from first principles involves a limit: dy/dx = limₕ→₀ [f(x+h) – iple proofs, understanding this limit deepens your grasp of what a derivative truly means.

从第一原理求导数的过程涉及极限：dy/dx = limₕ→₀ [f(x+h) – f(x)] / h。虽然 Edexcel 不再要求大量的第一原理证明，但理解这一极限能加深你对导数本质的认识。

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2. Power, Constant and Sum Rules | 幂函数法则、常数法则与和法则

ant), then dy/dx = 0. The sum rule: the derivative of a sum is the sum of the derivatives.

三个最基本的法则可以轻松处理多项式项。幂函数法则：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。常数法则：若 y = k（常数），则 dy/dx = 0。和法则：和的导数等于导数的和。

pplying the rules term by term: dy/dx = 20x⁴ − 9x² + 2. Notice the constant −7 disappears.

例如，求 y = 4x⁵ − 3x³ + 2x − 7 的导数。逐项应用法则：dy/dx = 20x⁴ − 9x² + 2。注意常数 −7 的导数为零。

Power rule generalisation: d/dx (a xⁿ) = a n xⁿ⁻¹

幂法则一般式：d/dx (a xⁿ) = a n xⁿ⁻¹

ppear in the denominator or as roots, always rewrite them in index form first: 1/x² = x⁻², √x = x½. Then apply the power rule directly.

如果 x 项出现在分母或根号中，务必先化为指数形式：1/x² = x⁻², √x = x½，然后直接使用幂法则。

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3. The Product Rule | 乘积法则

When two functions multiply together, y = u(x) ⋅ v(x), the product rule is essential: dy/dx = u (dv/dx) + v (du/dx). A useful mnemonic is "first times derivative of second plus second times derivative of first".

当两个函数相乘时，y = u(x) ⋅ v(x)，必须使用乘积法则：dy/dx = u (dv/dx) + v (du/dx)。记忆口诀：“前微后不微，加后微前不微”。

n du/dx = 3x², dv/dx = cos x. Hence dy/dx = x³ cos x + 3x² sin x = x² (x cos x + 3 sin x).

考虑 y = x³ sin x。设 u = x³，v = sin x，则 du/dx = 3x², dv/dx = cos x。因此 dy/dx = x³ cos x + 3x² sin x = x² (x cos x + 3 sin x)。

Always check if the original expression can be multiplied out before differentiating. If the product is simple, expanding may avoid the product rule, but when functions like ln x or sin x appear, you must use the product rule.

求导前经常可以先看看能否展开。如果乘积简单，展开可能规避乘积法则，但当含有 ln x 或 sin x 等函数时，必须使用乘积法则。

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4. The Quotient Rule | 商法则

/ v(x), the quotient rule states: dy/dx = [v (du/dx) − u (dv/dx)] / v². The order matters — numerator starts with v times derivative of u, not the other way around.

对于 y = u(x) / v(x)，商法则为：dy/dx = [v (du/dx) − u (dv/dx)] / v²。顺序很关键——分子是 v 乘 u 的导数，再减去 u 乘 v 的导数。

2x) − (x² + 1)(1)] / (x − 3)² = (2x² − 6x − x² − 1) / (x − 3)² = (x² − 6x − 1) / (x − 3)².

示例：求 y = (x² + 1) / (x − 3) 的导数。这里 u = x² + 1, v = x − 3, du/dx = 2x, dv/dx = 1。代入得：dy/dx = [(x − 3)(2x) − (x² + 1)(1)] / (x − 3)² = (x² − 6x − 1) / (x − 3)²。

Never reverse the subtraction, and remember to square v in the denominator. Leaving the answer in a simplified but factorised form is usually acceptable in Edexcel exams.

千万不要把减法顺序搞反，并记住分母要对 v 平方。在 Edexcel 考试中，将答案保留为化简但已因式分解的形式通常是可接受的。

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5. The Chain Rule | 链式法则

The chain rule handles composite functions — a function inside another. If y = f(g(x)), then dy/dx = f'(g(x)) ⋅ g'(x). Alternatively, using substitution u = g(x): dy/dx = (dy/du) × (du/dx).

链式法则处理复合函数——一个函数套在另一个内部。如果 y = f(g(x))，那么 dy/dx = f'(g(x)) ⋅ g'(x)。或者利用换元 u = g(x)：dy/dx = (dy/du) × (du/dx)。

(3x² + 5)⁷. Let u = 3x² + 5, then y = u⁷, dy/du = 7u⁶, du/dx = 6x. So dy/dx = 7(3x² + 5)⁶ ⋅ 6x = 42x(3x² + 5)⁶.

经典例子：y = (3x² + 5)⁷。令 u = 3x² + 5，则 y = u⁷, dy/du = 7u⁶, du/dx = 6x。因此 dy/dx = 7(3x² + 5)⁶ ⋅ 6x = 42x(3x² + 5)⁶。

ted operations. For instance, y = sin(2x+1) requires chain rule: derivative = 2 cos(2x+1).

链式法则常与乘积法则或商法则结合，因此要善于识别嵌套运算。例如，y = sin(2x+1) 需要用链式法则：导数为 2 cos(2x+1)。

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6. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分

The natural exponential function eˣ is unique: d/dx (eˣ) = eˣ. For a general exponential aˣ, the derivative is aˣ ln a. When the exponent is a function of x, apply the chain rule: d/dx (eᶠ⁽ˣ⁾) = f'(x) eᶠ⁽ˣ⁾.

自然指数函数 eˣ 非常独特：d/dx (eˣ) = eˣ。对于一般指数 aˣ，导数为 aˣ ln a。当指数是 x 的函数时，使用链式法则：d/dx (eᶠ⁽ˣ⁾) = f'(x) eᶠ⁽ˣ⁾。

Logarithmic derivatives: d/dx (ln x) = 1/x, for x > 0. More generally, d/dx (ln|f(x)|) = f'(x) / f(x). This pattern is extremely useful for integrating rational functions later.

对数函数的导数：d/dx (ln x) = 1/x (x > 0)。更一般地，d/dx (ln|f(x)|) = f'(x) / f(x)。这种形式在后续积分中非常有用。

Example: y = ln(cos x). Using the chain rule with u = cos x, dy/dx = (1/cos x) ⋅ (−sin x) = −tan x. Memorise these standard results; they appear frequently.

示例：y = ln(cos x)。用链式法则，令 u = cos x，则 dy/dx = (1/cos x) ⋅ (−sin x) = −tan x。请牢记这些标准结果，它们经常出现。

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

You must know the six basic trigonometric derivatives. The table below summarises them, with all angles measured in radians — differentiation in degrees would require an extra conversion factor.

你必须熟记六个基本三角函数的导数。下表总结了这些导数，所有角度均以弧度计量——若使用度数则需要额外的换算系数。

Function	Derivative	中文
sin x	cos x	正弦
cos x	−sin x	余弦
tan x	sec² x	正切
cot x	−cosec² x	余切
sec x	sec x tan x	正割
cosec x	−cosec x cot x	余割

d/dx sin(5x) = 5 cos(5x). Always multiply by the derivative of the inner linear function.

当角度为线性函数如 (ax + b) 时，需使用链式法则。例如，d/dx sin(5x) = 5 cos(5x)。一定要乘上内部线性函数的导数。

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8. Implicit Differentiation | 隐函数微分

th sides with respect to x, using the chain rule for any y-term: d/dx (yⁿ) = n yⁿ⁻¹ dy/dx.

许多方程无法显式解出 y。隐函数微分将 y 视为 x 的函数，对等式两边同时关于 x 求导，对含 y 的项使用链式法则：d/dx (yⁿ) = n yⁿ⁻¹ dy/dx。

−x/y. For more complicated implicit forms, gather all dy/dx terms on one side and factorise.

对圆 x² + y² = 25 求导，得 2x + 2y (dy/dx) = 0，因此 dy/dx = −x/y。对于更复杂的隐函数形式，将所有含 dy/dx 的项移到一边并提取公因子。

y − x²) / (y² − 2x). Implicit differentiation is indispensable for related rates problems.

示例：x³ + y³ = 6xy。求导：3x² + 3y² (dy/dx) = 6y + 6x (dy/dx)。整理得 dy/dx = (2y − x²) / (y² − 2x)。隐函数微分对于相关变化率问题不可或缺。

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9. Parametric Differentiation | 参数方程微分

y/dt) / (dx/dt), provided dx/dt ≠ 0. This formula comes from the chain rule: dy/dt = dy/dx ⋅ dx/dt.

当曲线用参数方程 x = f(t)、y = g(t) 定义时，梯度由 dy/dx = (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。该公式由链式法则推出：dy/dt = dy/dx ⋅ dx/dt。

t) = 1/t. To find the second derivative parametrically, use d²y/dx² = d(dy/dx)/dt ÷ dx/dt.

标准例子：x = t², y = 2t。则 dx/dt = 2t, dy/dt = 2，所以 dy/dx = 2 / (2t) = 1/t。为求参数方程的二阶导数，使用 d²y/dx² = d(dy/dx)/dt ÷ dx/dt。

Parametric equations often describe curves like the parabola or ellipse in a more tractable manner, and Edexcel exam questions will ask for tangents and normals in parametric form.

参数方程通常可以更简洁地描述抛物线、椭圆等曲线，Edexcel 考题常要求用参数形式求切线和法线。

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

lassifies it: positive → minimum, negative → maximum, zero → possible point of inflection.

二阶导数 d²y/dx² 反映梯度如何变化。在驻点处，dy/dx = 0。二阶导数的符号可对驻点分类：正 → 局部极小值，负 → 局部极大值，零 → 可能是拐点。

x. At x = 1, d²y/dx² = 6 > 0 → local minimum; at x = −1, d²y/dx² = −6 < 0 → local maximum.

对 y = x³ − 3x，dy/dx = 3x² − 3。令其为零，得 x = ±1。d²y/dx² = 6x。在 x = 1 处，d²y/dx² = 6 > 0 → 局部极小值；在 x = −1 处，d²y/dx² = −6 < 0 → 局部极大值。

If the second derivative test is inconclusive (zero), use the first derivative sign test either side of the point. Points of inflection occur where concavity changes, which can be confirmed by examining the second derivative sign change.

若二阶导数为零无法判定，可检查驻点左右两侧一阶导数的符号变化。拐点处凹向发生改变，可通过二阶导数符号变化来确认。

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11. Applications: Tangents, Normals, and Rates of Change | 应用：切线、法线与变化率

dy/dx evaluated at that point. The normal line is perpendicular, so its gradient is −1/m.

在特定点求出 dy/dx 后，切线方程为 y − y₁ = m (x − x₁)，其中 m 为该点处的导数值。法线垂直于切线，其梯度为 −1/m。

t: y − 3 = 4 (x − 1) → y = 4x − 1. Normal: gradient = −1/4, equation y − 3 = −1/4 (x − 1).

示例：求曲线 y = 2x² + 1 在 (1, 3) 处的切线。dy/dx = 4x，在 x=1 处 m = 4。切线：y − 3 = 4 (x − 1) → y = 4x − 1。法线：梯度 = −1/4，方程 y − 3 = −1/4 (x − 1)。

ps solve problems like the rate at which the area of a circle grows as its radius expands.

相关变化率需借助链式法则：若一个变量随时间变化并影响另一个，使用 dA/dt = dA/dr × dr/dt。这能解决诸如“圆面积随半径增大的变化率”等问题。

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12. Key Takeaways and Common Pitfalls | 核心要点与常见误区

Always rewrite roots and fractions using indices. Remember that eˣ is its own derivative, but the chain rule applies when the exponent is a function. Trigonometric derivatives require radian mode. With implicit and parametric differentiation, treat dy/dx as a symbolic entity to be solved for, never cancel differentials carelessly.

务必用指数形式重写根号和分式。记住 eˣ 的导数就是它本身，但当指数为函数时须用链式法则。三角函数微分必须在弧度制下进行。对于隐函数和参数方程，将 dy/dx 看作待求符号，切勿随意约去微分符号。

Many students lose marks by forgetting to multiply by the derivative of the inner function (chain rule), or by confusing product and quotient rules. Practice with mixed examples: differentiate x eˣ sin x or ln(x cos x) to build confidence.

许多学生因忘记乘上内层函数的导数（链式法则）或混淆乘积法则与商法则而失分。多做综合性练习，如对 x eˣ sin x 或 ln(x cos x) 求导，以增强信心。

Mastering these differentiation techniques not only secures high marks on the pure mathematics paper but also lays the foundation for integration techniques, differential equations, and further mechanics topics. A systematic approach and plenty of practice will make you fluent in calculus.

掌握这些微分技巧不仅能在纯数试卷上获得高分，也为积分技巧、微分方程以及进阶力学打下基础。系统方法与大量练习将使你熟练运用微积分。

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