📚 Edexcel A-Level Maths: Mastering Differentiation | 爱德思 A-Level 数学：掌握微分
Differentiation is one of the most important topics in the Edexcel A-Level Mathematics specification, appearing in Pure Mathematics papers and underpinning many applied problems in mechanics and statistics. This comprehensive guide covers the techniques, rules, and applications you need to master for exam success.
微分是爱德思 A-Level 数学大纲中最重要的主题之一，出现在纯数学试卷中，并为力学和统计中的许多应用问题奠定基础。本指南覆盖你在考试中取得成功所需掌握的技巧、法则和应用。
1. Differentiation from First Principles | 从第一原理求导
Differentiation from first principles uses the limit definition of the derivative. It is the foundation of all differentiation techniques and is regularly tested in Edexcel A-Level Maths.
从第一原理求导使用导数的极限定义。它是所有微分技巧的基础，也是爱德思 A-Level 数学中经常考查的内容。
f'(x) = limₕ→0 [f(x + h) – f(x)] / h
For example, if f(x) = x², then f'(x) = 2x after expanding and simplifying the expression f(x + h) – f(x).
例如，若 f(x) = x²，展开并化简 f(x + h) – f(x) 后可以得到 f'(x) = 2x。
2. Basic Rules: Power, Constant Multiple, Sum | 基本法则：幂函数、常数倍、和差
The power rule states that d/dx (xⁿ) = n xⁿ⁻¹. This rule works for any real n, including negative and fractional powers.
幂法则表明 d/dx (xⁿ) = n xⁿ⁻¹。该法则适用于任意实数 n，包括负指数和分数指数。
The constant multiple rule allows d/dx [k f(x)] = k f'(x), and the sum rule gives d/dx [f(x) + g(x)] = f'(x) + g'(x).
常数倍法则允许 d/dx [k f(x)] = k f'(x)，和差法则给出 d/dx [f(x) + g(x)] = f'(x) + g'(x)。
Always rewrite expressions like 1/x³ as x⁻³ and √x as x^½ before differentiating.
在求导前，务必先将 1/x³ 改写为 x⁻³，将 √x 改写为 x^½。
3. The Chain Rule | 链式法则
The chain rule is used for composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).
链式法则用于复合函数。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。
A common Edexcel application is differentiating expressions like (3x² + 5)⁴. The derivative is 4(3x² + 5)³ × 6x.
爱德思考试中常见应用是求 (3x² + 5)⁴ 这类表达式的导数。其导数为 4(3x² + 5)³ × 6x。
dy/dx = dy/du × du/dx
This alternative form is often useful when a substitution such as u = g(x) is made.
当使用 u = g(x) 进行换元时，这种替代形式通常非常有用。
4. The Product Rule | 乘积法则
For y = u(x) v(x), the product rule states dy/dx = u’ v + u v’.
对于 y = u(x) v(x)，乘积法则给出 dy/dx = u’ v + u v’。
Remember to set u and v clearly and differentiate each separately before substituting. Edexcel often asks for product rule with trigonometric or exponential functions.
请记住先清楚地设出 u 和 v，分别求导后再代入公式。爱德思常将乘积法则与三角函数或指数函数结合考查。
5. The Quotient Rule | 商法则
For y = u(x) / v(x), the quotient rule states dy/dx = (u’ v – u v’) / v².
对于 y = u(x) / v(x)，商法则给出 dy/dx = (u’ v – u v’) / v²。
Be careful with the subtraction sign – it is u’v minus uv’, not plus. The v² in the denominator must always be positive, even if v is negative.
注意减号——是 u’v 减去 uv’，不要写成加号。分母中的 v² 必须始终为正，即使 v 为负。
6. Differentiating Standard Functions | 标准函数的导数
You must know the derivatives of standard functions: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (ln x) = 1/x, and d/dx (eˣ) = eˣ.
你必须记住标准函数的导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (ln x) = 1/x，以及 d/dx (eˣ) = eˣ。
| Function | 函数 | Derivative | 导数 |
|---|---|
| sin kx | k cos kx |
| cos kx | -k sin kx |
| eᵏˣ | k eᵏˣ |
| ln kx | 1/x |
For ln kx, the derivative is still 1/x because ln kx = ln k + ln x and ln k is a constant.
对于 ln kx，其导数仍为 1/x，因为 ln kx = ln k + ln x，而 ln k 是常数。
7. Second and Higher Derivatives | 二阶及高阶导数
The second derivative, written d²y/dx² or f”(x), is obtained by differentiating the first derivative. It is used to determine the nature of stationary points.
二阶导数记作 d²y/dx² 或 f”(x)，由一阶导数再次求导得到。它用于判断驻点的性质。
Higher derivatives follow the same pattern, and Edexcel may ask for the third derivative in some questions involving differential equations or curve behaviour.
高阶导数遵循相同模式，爱德思在某些涉及微分方程或曲线行为的题目中可能要求三阶导数。
8. Applications: Tangents and Normals | 应用：切线与法线
The derivative at a point gives the gradient of the tangent. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal.
某点的导数给出切线的斜率。法线垂直于切线，因此其斜率是切线斜率的负倒数。
y – y₁ = m (x – x₁)
Equation of tangent at (x₁, y₁): y – y₁ = m (x – x₁), where m = f'(x₁). The normal has gradient -1/m.
切线方程在 (x₁, y₁)：y – y₁ = m (x – x₁)，其中 m = f'(x₁)。法线的斜率为 -1/m。
9. Applications: Stationary Points and Curve Sketching | 应用：驻点与曲线草图
Stationary points occur where f'(x) = 0. Use the second derivative to classify them: f”(x) > 0 for a minimum, f”(x) < 0 for a maximum, and f''(x) = 0 requires further investigation.
驻点出现在 f'(x) = 0 处。使用二阶导数判断类型：f”(x) > 0 为极小值，f”(x) < 0 为极大值，f''(x) = 0 时需进一步判断。
Sketching a curve involves finding intercepts, stationary points, and behaviour as x → ±∞. Always label key features clearly.
绘制曲线需要找出截距、驻点以及 x → ±∞ 时的行为。务必清晰标注关键特征。
10. Applications: Optimisation | 应用：优化问题
Optimisation problems ask you to maximise or minimise a quantity such as area, volume, or cost. Write the quantity as a function of one variable, differentiate, set f'(x) = 0, and verify the nature.
优化问题要求最大化或最小化面积、体积或成本等量。将该量写成单一变量的函数，求导，令 f'(x) = 0，并验证驻点性质。
Always check the physical domain of the variable and state your final answer with appropriate units.
务必检查变量的实际定义域，并在最终答案中注明适当的单位。
11. Rates of Change and Connected Rates | 变化率与相关变化率
If y changes with x and x changes with time t, then dy/dt = dy/dx × dx/dt. This is the chain rule applied to rates of change.
若 y 随 x 变化，且 x 随时间 t 变化，则 dy/dt = dy/dx × dx/dt。这是链式法则在变化率中的应用。
Typical Edexcel questions involve a balloon’s radius increasing or water flowing into a tank. Identify the given rate and the required rate before differentiating.
爱德思典型题目涉及气球半径增大或水流入水箱。在求导前先确定已知变化率和所求变化率。
12. Common Exam Mistakes and Tips | 常见考试错误与技巧
Always use brackets when differentiating negative powers or fractional powers. Do not forget to multiply by the derivative of the inner function when using the chain rule.
在求负指数或分数指数幂的导数时务必使用括号。使用链式法则时不要忘记乘以内层函数的导数。
Show clear working, especially when using product and quotient rules, because method marks are often available even if the final answer is wrong.
展示清晰的解题步骤，特别是在使用乘积法则和商法则时，因为即使最终答案错误，方法分通常仍然可以获得。
Practise past Edexcel papers to become confident in applying differentiation across mixed topics.
练习爱德思历年真题，以提高在混合题型中应用微分的信心。
Published by TutorHao | Edexcel A-Level Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导