📚 Differentiation Techniques and Applications | 微分技巧与应用
For Edexcel A-Level Mathematics, differentiation appears in Pure Paper 1 and Paper 2, and it also underpins mechanics and statistics contexts. Mastery of derivative rules, stationary points, tangents and optimisation is essential for top marks.
在 Edexcel A-Level 数学中，微分出现在 Pure Paper 1 和 Paper 2，并且是力学与统计情境的基础。掌握求导法则、驻点、切线和优化是获取高分的关键。
1. Differentiation from First Principles | 第一性原理求导
The gradient of a tangent is found as the limit of chord gradients. For a function f(x), the derivative f ‘(x) is defined by:
切线的斜率通过割线斜率的极限得到。对于函数 f(x)，导数 f ‘(x) 定义为：
f ‘(x) = lim (h→0) [f(x+h) − f(x)] / h
Edexcel frequently asks candidates to prove from first principles that the derivative of x² is 2x. Expand (x+h)², cancel x², divide by h, and then let h → 0.
Edexcel 经常要求考生从第一性原理证明 x² 的导数是 2x。展开 (x+h)²，消去 x²，除以 h，再令 h → 0。
2. Standard Derivatives and Basic Rules | 标准导数与基本法则
Learn the standard results for powers, exponentials, logarithms and trigonometric functions. These are used repeatedly in mixed questions.
熟记幂函数、指数函数、对数函数和三角函数的导数公式。这些公式在综合题中反复使用。
d/dx (xⁿ) = n xⁿ⁻¹ | d/dx (eˣ) = eˣ | d/dx (ln x) = 1/x | d/dx (sin x) = cos x | d/dx (cos x) = −sin x
The sum rule and constant multiple rule let you differentiate term by term, so each coefficient stays unchanged and the index drops by one.
和法则与常数倍法则允许你逐项求导，因此每一项的系数保持不变，指数减一。
3. Product Rule and Quotient Rule | 乘法法则与除法法则
Use the product rule when multiplying two functions. If y = u v, then:
两个函数相乘时使用乘法法则。若 y = u v，则：
dy/dx = u (dv/dx) + v (du/dx)
Use the quotient rule when dividing: y = u / v. Keep the denominator squared and the order v du minus u dv in the numerator.
相除时使用除法法则：y = u / v。分母保持平方，分子顺序为 v du 减 u dv。
dy/dx = [v (du/dx) − u (dv/dx)] / v²
4. Chain Rule and Parametric Differentiation | 链式法则与参数微分
The chain rule is used for composite functions. If y is a function of u and u is a function of x, then:
链式法则用于复合函数。如果 y 是 u 的函数且 u 是 x 的函数，则：
dy/dx = (dy/du) × (du/dx)
For parametric curves x = f(t), y = g(t), the gradient is:
对于参数曲线 x = f(t), y = g(t)，斜率为：
dy/dx = (dy/dt) ÷ (dx/dt)
This is essential for Edexcel questions on parametric equations, where you may also need d²y/dx² = d/dt (dy/dx) ÷ dx/dt.
这在 Edexcel 参数方程题目中必不可少，有时还要求 d²y/dx² = d/dt (dy/dx) ÷ dx/dt。
5. Tangents and Normals | 切线与法线
Once dy/dx is known at a point (x₁, y₁), the tangent gradient is m and the normal gradient is −1/m.
一旦求出点 (x₁, y₁) 处的 dy/dx，切线斜率为 m，法线斜率为 −1/m。
tangent: y − y₁ = m(x − x₁) | normal: y − y₁ = (−1/m)(x − x₁)
Substitute the original coordinates into the line equation; do not reuse the derivative formula as if it were the equation of the line.
将原始坐标代入直线方程；不要把导数公式本身误当成直线方程。
6. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where dy/dx = 0. To classify them, examine the sign of dy/dx on either side of the x-coordinate.
驻点出现在 dy/dx = 0 的位置。分类时要考察该 x 坐标左右两侧 dy/dx 的符号。
| dy/dx sign change | Type |
|---|---|
| + to − | Maximum |
| − to + | Minimum |
| same sign | Point of inflection |
If using the second derivative, f ”(x₀) > 0 gives a minimum, f ”(x₀) < 0 gives a maximum, and f ''(x₀) = 0 is inconclusive.
如果使用二阶导数，f ”(x₀) > 0 对应极小值，f ”(x₀) < 0 对应极大值，f ''(x₀) = 0 则无法判断。
7. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative measures how the gradient changes. A curve is convex (concave up) where f ”(x) > 0 and concave down where f ”(x) < 0.
二阶导数衡量斜率的变化。曲线在 f ”(x) > 0 时下凸（凹向上），在 f ”(x) < 0 时凹向下。
Points of inflection occur where the concavity changes and f ”(x) crosses zero. Check the sign change, not just f ”(x) = 0.
拐点出现在凹凸性改变且 f ”(x) 穿过零的位置。要检查符号变化，不能只看 f ”(x) = 0。
8. Optimisation Problems | 优化问题
Optimisation requires a quantity such as area, volume or cost written as a function of one variable. Differentiate, set dy/dx = 0, and solve for the variable.
优化问题时，需要把面积、体积或成本等量写成单一变量的函数。求导，令 dy/dx = 0，然后解出变量。
Always prove that a stationary value is a maximum or minimum. Use the sign test or second derivative, and answer the exact wording of the question.
始终要证明该驻值是最大值还是最小值。使用符号检验或二阶导数，并准确回答题目所问。
9. Connected Rates of Change | 相关变化率
Connected rates involve two or more variables changing with time. The chain rule links them, for example:
相关变化率涉及两个或多个随时间变化的变量。链式法则将它们联系起来，例如：
dV/dt = (dV/dr) × (dr/dt)
Edexcel often asks about a sphere’s volume V = (4/3)πr³ or a circle’s area A = πr². Substitute the given rate and the current radius carefully.
Edexcel 常考球体体积 V = (4/3)πr³ 或圆面积 A = πr²。代入给定的变化率和当前半径时要仔细。
10. Exam-Style Strategy and Common Errors | 考试策略与常见错误
Show every line of differentiation, even for simple terms, to gain method marks. Use exact values such as √2 and π unless the question asks otherwise.
即使是简单项，也要写出每一步求导过程，以获得方法分。除非题目要求，否则使用 √2、π 等精确值。
Common errors include forgetting the chain rule for eᵏˣ or sin(kx), mishandling the quotient numerator order, and failing to classify stationary points.
常见错误包括忘记对 eᵏˣ 或 sin(kx) 使用链式法则、弄错除法法则分子的顺序，以及没有对驻点进行分类。
In word problems, define variables clearly, state the domain, and interpret your mathematical answer in context.
在应用题中，清楚地定义变量，说明取值范围，并将数学答案解释回实际情境。
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