IGCSE OCR Maths: Partial Differentiation Essentials | IGCSE OCR 数学：偏微分考点精讲

📚 IGCSE OCR Maths: Partial Differentiation Essentials | IGCSE OCR 数学：偏微分考点精讲

This article covers the key differentiation topics for IGCSE OCR Mathematics. Although ‘partial differentiation’ is not part of the IGCSE syllabus (it appears in advanced courses), we thoroughly explain ordinary differentiation, which is essential for the exam, and touch on partial differentiation as a bridge to future study.

本文梳理 IGCSE OCR 数学中微分的关键考点。虽然 ‘偏微分’ 不在 IGCSE 大纲内（属于进阶课程内容），但我们先全面讲解考试必备的常微分，再简要介绍偏微分概念，为深阶学习搭桥铺路。

1. What Is Differentiation? | 什么是微分？

Differentiation is the process of finding the derivative of a function. The derivative measures how a function’s output changes as its input changes, giving the gradient of the curve at any point.

微分是求函数导数的过程。导数衡量函数输出随输入变化的快慢，描述曲线上任意一点的切线斜率。

In IGCSE, we deal with functions of a single variable, usually y = f(x), and the derivative is denoted by f'(x) or dy/dx.

在 IGCSE 中，我们处理单变量函数，通常为 y = f(x)，导数记作 f'(x) 或 dy/dx。

2. Gradient of a Curve | 曲线的梯度

The gradient of a straight line is constant, but for a curve, the gradient varies from point to point. The derivative gives the instantaneous rate of change, i.e., the gradient of the tangent at a specific x-value.

直线梯度恒定，但曲线的梯度随点变化。导数表示瞬时变化率，即特定 x 值处切线的梯度。

For example, for y = x², the derivative dy/dx = 2x, so the gradient at x = 3 is 6.

例如，对于 y = x²，导数 dy/dx = 2x，因此在 x = 3 处的梯度为 6。

3. Differentiation from First Principles | 从第一原理求导

The formal definition of the derivative is the limit of the difference quotient: f'(x) = limₕ→₀ (f(x+h) – f(x)) / h. This is called differentiation from first principles.

导数的正式定义为差商的极限：f'(x) = limₕ→₀ (f(x+h) – f(x)) / h。这称为从第一原理求导。

While OCR IGCSE does not routinely require finding derivatives from first principles, understanding this concept deepens your grasp of what a derivative truly represents.

尽管 OCR IGCSE 不常要求从第一原理求导，但理解这一概念能加深你对导数本质的认识。

4. The Power Rule | 幂法则

For functions of the form y = xⁿ, the derivative is dy/dx = n xⁿ⁻¹. This is the most commonly used differentiation rule at IGCSE.

对于 y = xⁿ 形式的函数，其导数为 dy/dx = n xⁿ⁻¹。这是 IGCSE 中最常用的求导法则。

Examples: derivative of x³ is 3x²; derivative of x¹⁄₂ (√x) is ½ x⁻¹⁄₂ = 1/(2√x); derivative of constant is 0.

示例：x³ 的导数为 3x²；x¹⁄₂ (√x) 的导数为 ½ x⁻¹⁄₂ = 1/(2√x)；常数的导数为 0。

5. Sum and Difference Rule | 和差法则

The derivative of a sum (or difference) of two functions is the sum (or difference) of their derivatives: if y = u(x) ± v(x), then dy/dx = du/dx ± dv/dx.

两个函数之和（或差）的导数等于各自导数之和（或差）：若 y = u(x) ± v(x)，则 dy/dx = du/dx ± dv/dx。

This allows us to differentiate polynomial terms term by term. For y = 4x³ + 2x² – x + 5, the derivative is dy/dx = 12x² + 4x – 1.

这让我们能够逐项对多项式求导。例如 y = 4x³ + 2x² – x + 5，导数为 dy/dx = 12x² + 4x – 1。

6. Constant Multiple Rule | 常数倍法则

If y = k · f(x), where k is a constant, then dy/dx = k · f'(x). The constant factor simply stays with the derivative.

若 y = k · f(x)，k 为常数，则 dy/dx = k · f'(x)。常数因子可以提到导数符号外。

For instance, the derivative of 10x⁴ is 10·4x³ = 40x³.

例如，10x⁴ 的导数为 10·4x³ = 40x³。

7. Tangents to Curves | 曲线的切线

To find the equation of a tangent at a point (x₁, y₁) on a curve y = f(x): calculate the gradient m = f'(x₁), then use the line equation y – y₁ = m(x – x₁).

求曲线 y = f(x) 上点 (x₁, y₁) 处的切线方程：先计算梯度 m = f'(x₁)，再利用直线方程 y – y₁ = m(x – x₁)。

Example: for y = x² + 3x at x = 1, y = 1 + 3 = 4. f'(x) = 2x + 3, so m = 5. Tangent equation: y – 4 = 5(x – 1), i.e., y = 5x – 1.

示例：曲线 y = x² + 3x 在 x = 1 处，y = 4。f'(x) = 2x + 3，故 m = 5。切线方程：y – 4 = 5(x – 1)，即 y = 5x – 1。

8. Normals to Curves | 曲线的法线

A normal is perpendicular to the tangent. Its gradient m_normal = -1 / m_tangent, provided m_tangent ≠ 0. The equation is found similarly using the point-slope form.

法线垂直于切线。其梯度 m_normal = -1 / m_tangent（切线梯度不为 0 时）。同样使用点斜式求方程。

For the previous example, m_tangent = 5, so m_normal = -1/5. Normal at (1,4): y – 4 = -1/5 (x – 1) => y = -1/5 x + 21/5.

沿用前例，切线梯度为 5，则法线梯度为 -1/5。在 (1,4) 的法线：y – 4 = -1/5 (x – 1)，即 y = -1/5 x + 21/5。

9. Second Derivative | 二阶导数

The second derivative, written d²y/dx² or f”(x), is the derivative of the first derivative. It measures the rate of change of the gradient, indicating the concavity of the function.

二阶导数，写作 d²y/dx² 或 f”(x)，是一阶导数的导数。它衡量梯度的变化率，显示函数的凹凸性。

For y = 3x⁴ – 2x² + x, f'(x) = 12x³ – 4x + 1, so f”(x) = 36x² – 4. The second derivative is used to determine the nature of stationary points.

对于 y = 3x⁴ – 2x² + x，f'(x) = 12x³ – 4x + 1，故 f”(x) = 36x² – 4。二阶导数常用于判断驻点性质。

10. Stationary Points: Maxima and Minima | 驻点：极大值与极小值

Stationary points occur where f'(x) = 0. To classify them, use the second derivative test: if f”(x) > 0, it’s a local minimum; if f”(x) < 0, a local maximum; if f''(x) = 0, further investigation is needed.

驻点满足 f'(x) = 0。使用二阶导数判别：若 f”(x) > 0，为局部极小值；若 f”(x) < 0，为局部极大值；若 f''(x) = 0，需进一步分析。

Example: y = x³ – 3x. f'(x) = 3x² – 3 = 0 gives x = ±1. f”(x) = 6x, so at x = 1, f”(1) = 6 > 0 (minimum); at x = -1, f”(-1) = -6 < 0 (maximum).

例如：y = x³ – 3x。f'(x) = 3x² – 3 = 0 得 x = ±1。f”(x) = 6x，当 x = 1，f”(1) = 6 > 0 (极小)；当 x = -1，f”(-1) = -6 < 0 (极大)。

11. Kinematics Applications | 运动学应用

In straight-line motion, displacement s, velocity v, and acceleration a are related by differentiation: v = ds/dt and a = dv/dt = d²s/dt².

在直线运动中，位移 s、速度 v 和加速度 a 通过求导关联：v = ds/dt，a = dv/dt = d²s/dt²。

OCR IGCSE questions often provide a displacement formula s = f(t) and ask for velocity or acceleration at a given time. Remember that the gradient of a displacement-time graph gives velocity.

OCR IGCSE 考题常给出位移公式 s = f(t)，要求某时刻的速度或加速度。牢记位移-时间图的梯度就是速度。

12. Beyond IGCSE: Partial Differentiation | 超越 IGCSE：偏微分简介

Partial differentiation extends the idea to functions of several variables, e.g., u = f(x,y). The partial derivative ∂u/∂x treats y as constant and differentiates with respect to x. This is not examined at IGCSE but forms the foundation for advanced calculus in A-level Further Mathematics and university STEM courses.

偏微分将求导概念推广到多变量函数，如 u = f(x,y)。偏导数 ∂u/∂x 将 y 视为常数对 x 求导。这一内容虽不属 IGCSE 考试，却是 A-level 进阶数学及大学理工科高等微积分的重要基石。

If you have mastered ordinary differentiation, partial differentiation is a natural next step—just remember to keep the other variables fixed while you differentiate with respect to one.

掌握常微分后，偏微分是顺理成章的下一步——只需记住在对某一变量求导时，将其他变量当作常数即可。

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IGCSE OCR Maths: Partial Differentiation Essentials | IGCSE OCR 数学：偏微分 考点精讲