Mastering A-Level Edexcel Pure Maths: Differentiation Techniques | A-Level Edexcel 纯数学：微分技巧精讲

📚 Mastering A-Level Edexcel Pure Maths: Differentiation Techniques | A-Level Edexcel 纯数学：微分技巧精讲

Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. From first principles to parametric and implicit methods, a clear understanding of derivatives underpins many applied problems, including rates of change, optimisation and curve sketching. This revision guide breaks down the key techniques and common pitfalls to help you build confidence and accuracy.

微分是 Edexcel A-Level 纯数学中考查频率最高的主题之一。从第一性原理到参数方程与隐函数求导，对导数的清晰理解是许多应用问题的基础，包括变化率、最优化和曲线草图。本复习指南分解关键技巧和常见错误，帮助你建立信心并提高准确度。

1. First Principles and the Definition of the Derivative | 第一性原理与导数定义

The derivative of a function measures its instantaneous rate of change. For a function f(x), the derivative f'(x) is defined by the limit f'(x) = lim (h→0) [f(x+h)-f(x)]/h. Although Edexcel does not require a full epsilon-delta proof, you must be able to apply the limit definition to simple polynomials and interpret the gradient of a tangent.

函数的导数衡量其瞬时变化率。对于函数 f(x)，导数 f'(x) 由极限定义 f'(x) = lim (h→0) [f(x+h)-f(x)]/h。虽然 Edexcel 不要求完整的ε-δ证明，但你必须能够对简单多项式运用极限定义，并解释切线的斜率。

For example, to differentiate f(x) = x² from first principles, expand f(x+h) = (x+h)² = x² + 2xh + h². Then the expression [f(x+h)-f(x)]/h becomes (2xh + h²)/h = 2x + h, and as h tends to 0 the limit is 2x.

例如，用第一性原理求 f(x) = x² 的导数，展开 f(x+h) = (x+h)² = x² + 2xh + h²。此时 [f(x+h)-f(x)]/h 变为 (2xh + h²)/h = 2x + h，当 h 趋于 0 时极限为 2x。

f'(x) = lim (h→0) [f(x+h)-f(x)]/h

2. Standard Derivatives and the Power Rule | 标准导数与幂法则

The power rule is the foundation for most differentiation in A-Level Pure Maths. If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹, where n can be any real constant. This rule also applies to fractional and negative powers such as x^(1/2), x⁻¹ and x^(3/2), so rewriting roots and reciprocals as powers is essential before differentiating.

幂法则是 A-Level 纯数学中大多数微分运算的基础。若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 可为任意实数常数。该法则同样适用于分数幂和负幂，例如 x^(1/2)、x⁻¹ 和 x^(3/2)，因此在求导前将根式和倒数改写为幂的形式至关重要。

d/dx (xⁿ) = n xⁿ⁻¹

You should memorise the standard derivatives: sin x differentiates to cos x, cos x to -sin x, tan x to sec² x, eˣ to eˣ, ln x to 1/x, aˣ to aˣ ln a, and any constant k to 0.

你应熟记标准导数：sin x 的导数为 cos x，cos x 的导数为 -sin x，tan x 的导数为 sec² x，eˣ 的导数为 eˣ，ln x 的导数为 1/x，aˣ 的导数为 aˣ ln a，任意常数 k 的导数为 0。

3. Chain Rule | 链式法则

The chain rule is used when differentiating a function of a function. If y is a function of u and u is a function of x, then dy/dx = dy/du × du/dx. It is particularly useful for composite expressions such as powers of brackets, exponential and trigonometric functions containing linear or nonlinear inner functions.

链式法则用于对复合函数求导。若 y 是 u 的函数且 u 是 x 的函数，则 dy/dx = dy/du × du/dx。它特别适用于括号幂、指数函数和三角函数内部含有线性或非线性函数的复合表达式。

For example, if y = (3x² + 5)⁴, set u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.

例如，若 y = (3x² + 5)⁴，设 u = 3x² + 5。则 dy/du = 4u³ 且 du/dx = 6x，因此 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。

dy/dx = dy/du × du/dx

4. Product Rule | 乘积法则

When differentiating a product of two functions of x, use the product rule. If y = u v, then dy/dx = u’ v + u v’. You should identify u and v clearly, differentiate each separately, and then substitute into the formula. Keeping the order u’ v + u v’ can reduce sign errors.

当对两个关于 x 的函数之积求导时，使用乘积法则。若 y = u v，则 dy/dx = u’ v + u v’。你应清晰地确定 u 和 v，分别求导，然后代入公式。保持 u’ v + u v’ 的顺序可以减少符号错误。

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

例如，若 y = x² sin x，设 u = x² 且 v = sin x。则 u’ = 2x 且 v’ = cos x，因此 dy/dx = 2x sin x + x² cos x。

d/dx (u v) = u’ v + u v’

5. Quotient Rule | 商法则

The quotient rule is required for differentiating a function expressed as one function divided by another. If y = u/v, then dy/dx = (u’ v – u v’)/v². The numerator order is critical: it must be u’ v minus u v’, not the reverse. A common exam error is reversing this order or forgetting to square the denominator.

商法则用于对一个函数除以另一个函数的形式求导。若 y = u/v，则 dy/dx = (u’ v – u v’)/v²。分子的顺序至关重要：必须是 u’ v 减去 u v’，不能颠倒。一个常见的考试错误就是颠倒该顺序或忘记对分母平方。

For example, if y = (x² + 1)/(x – 2), set u = x² + 1 and v = x – 2. Then u’ = 2x and v’ = 1, so dy/dx = [2x(x – 2) – (x² + 1)(1)]/(x – 2)² = (2x² – 4x – x² – 1)/(x – 2)² = (x² – 4x – 1)/(x – 2)².

例如，若 y = (x² + 1)/(x – 2)，设 u = x² + 1 且 v = x – 2。则 u’ = 2x 且 v’ = 1，因此 dy/dx = [2x(x – 2) – (x² + 1)(1)]/(x – 2)² = (2x² – 4x – x² – 1)/(x – 2)² = (x² – 4x – 1)/(x – 2)²。

d/dx (u/v) = (u’ v – u v’)/v²

6. Implicit Differentiation | 隐函数求导

Implicit differentiation is used when y is not explicitly isolated, for example in equations such as x² + y² = 25 or x³ + xy + y² = 7. Differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to y terms, so the derivative of y² is 2y dy/dx and the derivative of xy requires the product rule.

隐函数求导用于 y 未被显式分离的情况，例如 x² + y² = 25 或 x³ + xy + y² = 7 这类方程。对方程两边关于 x 求导，将 y 视为 x 的函数并对 y 的项应用链式法则，因此 y² 的导数为 2y dy/dx，而 xy 的导数需要使用乘积法则。

For example, differentiating x² + y² = 25 gives 2x + 2y dy/dx = 0, which rearranges to dy/dx = -x/y. This shows that the gradient of a circle tangent depends on both coordinates.

例如，对 x² + y² = 25 求导得到 2x + 2y dy/dx = 0，整理得 dy/dx = -x/y。这表明圆的切线斜率同时取决于两个坐标。

d/dx (y²) = 2y dy/dx

7. Parametric Differentiation | 参数方程求导

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: dy/dx = (dy/dt)/(dx/dt), provided dx/dt is not zero. This allows the tangent gradient to be calculated without eliminating the parameter.

当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，斜率 dy/dx 可通过 dy/dt 除以 dx/dt 求得：dy/dx = (dy/dt)/(dx/dt)，前提是 dx/dt 不为零。这样无需消去参数即可计算切线斜率。

For example, if x = t² and y = t³, then dx/dt = 2t and dy/dt = 3t², so dy/dx = 3t²/2t = 3t/2. The second derivative d²y/dx² is obtained by differentiating dy/dx with respect to x, so d²y/dx² = d/dt (dy/dx) ÷ dx/dt.

例如，若 x = t² 且 y = t³，则 dx/dt = 2t 且 dy/dt = 3t²，因此 dy/dx = 3t²/2t = 3t/2。二阶导数 d²y/dx² 可通过对 dy/dx 关于 x 再求导得到，即 d²y/dx² = d/dt (dy/dx) ÷ dx/dt。

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

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

The second derivative f”(x) gives the rate of change of the gradient and indicates the curvature of a function. If f”(x) > 0 on an interval, the curve is convex, meaning it bends upwards and a tangent lies below the graph. If f”(x) < 0, the curve is concave, bending downwards.

二阶导数 f”(x) 表示斜率的变化率，并反映函数的弯曲方向。若在某个区间内 f”(x) > 0，则曲线是凸的，即向上弯曲且切线位于图像下方。若 f”(x) < 0，则曲线是凹的，向下弯曲。

A point where the concavity changes sign is called a point of inflection. At such a point, the second derivative is zero or undefined, but you must check that f”(x) actually changes sign on either side.

凹凸性发生符号变化的点称为拐点。在拐点处，二阶导数为零或不存在，但你必须检验 f”(x) 在该点两侧是否确实变号。

f”(x) > 0 ⇒ convex, f”(x) < 0 ⇒ concave

9. Rates of Change and Connected Rates | 变化率与关联变化率

Differentiation is widely used to model rates of change. If a quantity A depends on radius r, and r depends on time t, then the chain rule connects the rates: dA/dt = dA/dr × dr/dt. This is called a connected rates problem and often appears in Edexcel exam questions involving circles, spheres, cones or other geometric shapes.

微分广泛用于建立变化率模型。若量 A 依赖于半径 r，而 r 又依赖于时间 t，则链式法则将各变化率联系起来：dA/dt = dA/dr × dr/dt。这类问题称为关联变化率，经常出现在 Edexcel 考试中，涉及圆、球、锥或其他几何图形。

For example, the area of a circle is A = π r². Differentiating with respect to r gives dA/dr = 2π r. If the radius increases at 3 cm/s, the rate of change of area is dA/dt = 2π r × 3 = 6π r cm²/s.

例如，圆的面积为 A = π r²。对 r 求导得 dA/dr = 2π r。若半径以 3 cm/s 的速度增加，则面积的变化率为 dA/dt = 2π r × 3 = 6π r cm²/s。

dA/dt = dA/dr × dr/dt

10. Optimisation and Curve Sketching | 最优化与曲线草图

To find stationary points, set the first derivative equal to zero and solve for x. The stationary point is a maximum if f”(x) < 0 and a minimum if f''(x) > 0. If the second derivative is zero, use a sign test of f'(x) on either side or examine the second derivative further.

求驻点时，令一阶导数等于零并解出 x。若 f”(x) < 0，该驻点为极大值点；若 f''(x) > 0，则为极小值点。若二阶导数为零，则使用一阶导数在两侧的符号检验，或进一步考察二阶导数。

Optimisation problems require you to express one quantity in terms of one variable, differentiate, set the derivative to zero, and verify the nature of the stationary point. Always check the domain of the original problem, including endpoints if the interval is closed.

最优化问题要求你将某个量用一个变量表示，对其求导，令导数为零，并检验驻点的性质。务必检查原问题的定义域，若区间是闭区间还应包括端点。

For example, if V = x²(10 – x) for 0 < x < 10, then V' = 20x - 3x². Setting V' = 0 gives x = 0 or x = 20/3, and the second derivative test identifies x = 20/3 as a maximum.

例如，若 V = x²(10 – x) 且 0 < x < 10，则 V' = 20x - 3x²。令 V' = 0 得 x = 0 或 x = 20/3，二阶导数检验可确定 x = 20/3 为极大值点。

11. Common Errors and Exam Tips | 常见错误与应试技巧

Students often lose marks through small but preventable mistakes. When using the chain rule, do not forget to multiply by the derivative of the inner function. In the quotient rule, keep the numerator as u’ v – u v’ and remember to divide by v², not v.

学生常常因细小但可避免的错误而失分。使用链式法则时，不要忘记乘以内层函数的导数。使用商法则时，分子应保持为 u’ v – u v’，并记住要除以 v² 而非 v。

Other common pitfalls include confusing the derivative of sin x with the derivative of cos x, failing to simplify negative and fractional powers before differentiating, and mixing up dy/dx with dx/dy in related rates. Always write clear notation and show each step in the exam.

其他常见错误包括将 sin x 的导数与 cos x 的导数混淆、在求导前未能化简负幂和分数幂、以及在关联变化率中混淆 dy/dx 与 dx/dy。考试中务必书写清晰符号并展示每一步过程。

Use standard derivatives from memory, but double-check signs.
Check that your derivative is dimensionally consistent in applied questions.
Simplify answers before differentiating where possible.
Always state dy/dx explicitly when answering gradient questions.

熟记标准导数，但仔细检查符号。
在应用问题中检查导数的量纲是否一致。
尽可能先化简再求导。
回答斜率问题时务必明确写出 dy/dx。

12. Summary and Practice Checklist | 总结与练习清单

Mastering differentiation for Edexcel A-Level Pure Maths requires fluent application of the power, chain, product, quotient, implicit and parametric rules, together with a solid understanding of second derivatives and rates of change. Regular practice with past paper questions is the best way to secure high marks.

掌握 Edexcel A-Level 纯数学的微分内容，需要熟练运用幂法则、链式法则、乘积法则、商法则、隐函数求导和参数方程求导，并扎实理解二阶导数与变化率。定期练习历年真题是确保高分的最佳途径。

Use the following checklist before your exam: can you differentiate from first principles? Can you recognise when to use each rule? Can you find stationary points and classify them? Can you solve connected rates problems? If yes, you are well prepared.

考试前请使用以下清单：你能从第一性原理求导吗？你能识别何时使用每种法则吗？你能求出驻点并进行分类吗？你能解决关联变化率问题吗？如果能，你已经准备充分。

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Mastering A-Level Edexcel Pure Maths: Differentiation Techniques | A-Level Edexcel 纯数学：微分技巧精讲