📚 Partial Differentiation | 偏微分 考点精讲
In IGCSE Edexcel Mathematics, differentiation is a central topic that equips you to find rates of change and gradients of curves. Although ‘partial differentiation’ is an advanced concept typically reserved for A-Level Further Mathematics, grasping its fundamentals by linking back to single-variable differentiation can give you a head start. This comprehensive revision guide will first solidify your IGCSE differentiation skills and then introduce partial derivatives in a gentle, step-by-step manner.
在IGCSE Edexcel数学中，微分是一个核心主题，它让你能够求出变化率和曲线的斜率。虽然“偏微分”是一个更高级的概念，通常会留在A-Level进阶数学中学习，但通过联系单变量微分来掌握其基础可以让你领先一步。这份全面的复习指南将首先巩固你的IGCSE微分技能，然后以温和、循序渐进的方式介绍偏导数。
1. What is Differentiation? | 什么是微分？
Differentiation measures how a function changes as its input changes. For a curve y = f(x), the derivative f’ (x) or dy/dx gives the slope of the tangent at any point. It is the limit of the average rate of change Δy/Δx as Δx approaches zero.
微分衡量的是函数随输入变化而变化的程度。对于曲线 y = f(x)，导数 f'(x) 或 dy/dx 给出了任意点切线的斜率。它是当 Δx 趋近于零时，平均变化率 Δy/Δx 的极限。
2. The Power Rule | 幂法则
The most fundamental rule for differentiating polynomials is the power rule: if y = xn, then dy/dx = n xn–1. This applies for any real power n.
微分多项式最基本的法则是幂法则：如果 y = xn，那么 dy/dx = n xn–1。这适用于任意实数指数 n。
For example:
y = x3 → dy/dx = 3x2
y = 5x2 → dy/dx = 10x
y = √x = x½ → dy/dx = ½ x–½ = 1/(2√x)
例如：
y = x3 → dy/dx = 3x2
y = 5x2 → dy/dx = 10x
y = √x = x½ → dy/dx = ½ x–½ = 1/(2√x)
3. Derivatives of Common Functions | 常见函数的导数
Besides polynomials, IGCSE Edexcel often tests the derivatives of trigonometric, exponential, and logarithmic functions. Here is a quick reference table:
除了多项式，IGCSE Edexcel 常考三角函数、指数函数和对数函数的导数。下面是一个速查表：
| Function f(x) | Derivative f'(x) |
|---|---|
| sin x | cos x |
| cos x | –sin x |
| ex | ex |
| ln x | 1/x |
These should be memorised, as they appear frequently in gradient and rate-of-change problems.
这些导数应当牢记，因为它们经常出现在斜率和变化率问题中。
4. Tangents and Normals | 切线与法线
With the derivative, you can find the equation of the tangent at a point (x₁, y₁). The gradient of the tangent is m = dy/dx evaluated at that point. The normal is perpendicular to the tangent, so its gradient is –1/m.
有了导数，你就能求出点 (x₁, y₁) 处的切线方程。切线的斜率就是该点处的导数 m = dy/dx。法线与切线垂直，因此它的斜率为 –1/m。
Example: For y = x2 + 3x at x = 1, y = 4. The derivative is 2x + 3, so m = 5. Tangent: y – 4 = 5(x – 1). Normal: y – 4 = –1/5 (x – 1).
例子：对于 y = x2 + 3x，在 x=1 处，y=4。导数为 2x+3，因此 m=5。切线：y – 4 = 5(x – 1)。法线：y – 4 = –1/5 (x – 1)。
5. Second Derivative and Stationary Points | 二阶导数与驻点
The second derivative, d2y/dx2, tells you about the concavity of a graph. Setting the first derivative to zero gives stationary points (turning points). If f”(x) > 0, the point is a local minimum; if f”(x) < 0, it is a local maximum. When f''(x) = 0, further investigation is needed (possible point of inflection).
二阶导数 d2y/dx2 反映了图形的凹凸性。令一阶导数为零可求得驻点（转折点）。若 f”(x) > 0，该点为局部极小值；若 f”(x) < 0，则为局部极大值。当 f''(x) = 0 时，需要进一步判断（可能是拐点）。
For example, y = x3 – 3x has dy/dx = 3x2 – 3 = 0 at x = ±1. The second derivative is 6x: at x = 1, f”(1) = 6 > 0 → minimum; at x = –1, f”(–1) = –6 < 0 → maximum.
例如，y = x3 – 3x 的导数 dy/dx = 3x2 – 3，在 x = ±1 处为零。二阶导数为 6x：在 x=1 处，f”(1)=6>0，为极小值；在 x=–1 处，f”(–1)=–6<0，为极大值。
6. Application in Kinematics | 运动学中的应用
In kinematics, displacement s, velocity v, and acceleration a are linked by differentiation. If s(t) is the position, then v = ds/dt and a = dv/dt = d2s/dt2. This is a key IGCSE topic, especially in context of motion with constant or variable acceleration.
在运动学中，位移 s、速度 v 和加速度 a 通过微分联系起来。如果 s(t) 是位置，那么 v = ds/dt，a = dv/dt = d2s/dt2。这是 IGCSE 的一个重要考点，尤其在常加速度或变加速度的运动情境中。
For instance, if s = t3 – 2t2, then v = 3t2 – 4t and a = 6t – 4. You can find when the particle is at rest (v = 0) or the acceleration at a specific time.
例如，若 s = t3 – 2t2，则 v = 3t2 – 4t，a = 6t – 4。你可以求出质点何时静止（v=0）或某一时刻的加速度。
7. Functions of More Than One Variable | 多变量函数Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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