GCSE CIE Maths: Differentiation Revision Notes | GCSE CIE 数学：微分考点精讲

📚 GCSE CIE Maths: Differentiation Revision Notes | GCSE CIE 数学：微分考点精讲

Welcome to this comprehensive revision guide on Differentiation for the CIE IGCSE Mathematics (Extended) syllabus. Differentiation is a fundamental tool in calculus that allows you to find the gradient of a curve at any point, determine rates of change, and solve real-world optimisation problems. Mastering differentiation is essential for success in the exam and provides a strong foundation for further mathematics.

欢迎阅读这篇针对 CIE IGCSE 数学（扩展）的微分考点精讲。微分是微积分中的基本工具，它能让我们求出曲线上任意一点的梯度，确定变化率，并解决现实中的优化问题。掌握微分对考试成功至关重要，也为更高层次数学打好基础。

1. What is Differentiation? | 什么是微分？

Differentiation is the process of finding the rate at which one quantity changes with respect to another. The result is called the derivative, written as dy/dx or f'(x). Geometrically, the derivative gives the gradient of the tangent to a curve at a specific point.

微分是求一个量相对于另一个量变化率的过程，其结果称为导数，记作 dy/dx 或 f'(x)。在几何上，导数给出了曲线在某一点处切线的斜率。

The derivative of a constant is zero because a constant does not change. For a linear function y = mx + c, the derivative is simply the gradient m.

常数的导数为零，因为常数不变。对于线性函数 y = mx + c，导数就是斜率 m。

When we differentiate a function, we produce a new function that gives the gradient at any x-value. This is why the derivative is often called the gradient function.

对一个函数求导，会得到一个新的函数，该函数可给出任意 x 值处的梯度，因此导数常被称为梯度函数。

2. Differentiating xⁿ | xⁿ 的求导法则

For any real number n, the derivative of y = xⁿ is given by:

对于任意实数 n，y = xⁿ 的导数为：

dy/dx = n xⁿ⁻¹

To differentiate a power term, multiply by the power and then reduce the power by one. For example, if y = x⁵, then dy/dx = 5x⁴.

对幂函数求导时，乘以幂指数，再将幂指数减一。例如，若 y = x⁵，则 dy/dx = 5x⁴。

If there is a constant coefficient, it stays as a multiplier: for y = kxⁿ, the derivative is dy/dx = k n xⁿ⁻¹.

若带有常数系数，常数系数保留为乘数：对于 y = kxⁿ，导数为 dy/dx = k n xⁿ⁻¹。

3. Negative and Fractional Powers | 负指数与分数指数

The power rule works exactly the same way when n is negative or a fraction. Remember to rewrite roots and reciprocals as powers: 1/x² = x⁻², and √x = x^(½).

当 n 为负数或分数时，幂法则完全适用。记住将根式和倒数改写为幂形式：1/x² = x⁻²，√x = x^(½)。

For y = x⁻³, dy/dx = -3x⁻⁴.

对于 y = x⁻³，dy/dx = -3x⁻⁴。
For y = 1/x = x⁻¹, dy/dx = -x⁻² = -1/x².

对于 y = 1/x = x⁻¹，dy/dx = -x⁻² = -1/x²。
For y = √x = x^(½), dy/dx = (½)x^(-½) = 1/(2√x).

对于 y = √x = x^(½)，dy/dx = (½)x^(-½) = 1/(2√x)。
For y = 5/x³ = 5x⁻³, dy/dx = -15x⁻⁴.

对于 y = 5/x³ = 5x⁻³，dy/dx = -15x⁻⁴。

4. Differentiating Sums and Differences | 和与差的导数

You can differentiate a polynomial term by term. If y = u + v, then dy/dx = du/dx + dv/dx, where u and v are functions of x. The same holds for subtraction.

多项式可以逐项求导。若 y = u + v，则 dy/dx = du/dx + dv/dx，其中 u 和 v 都是 x 的函数。减法同理。

For example, y = 4x³ – 2x² + 7x – 9. Differentiating each term: dy/dx = 12x² – 4x + 7. The derivative of the constant -9 is zero.

例如 y = 4x³ – 2x² + 7x – 9。逐项求导：dy/dx = 12x² – 4x + 7。常数 -9 的导数为零。

This rule makes it easy to handle any polynomial expression by breaking it into simple power terms.

该法则使得处理任意多项式都很容易，只需将其分解为简单的幂次项。

5. Tangents and Normals | 切线与法线

To find the equation of a tangent at a point (x₁, y₁) on a curve y = f(x):

求曲线 y = f(x) 上点 (x₁, y₁) 处切线方程的步骤：

Calculate the gradient m = f'(x₁) by differentiating and substituting x₁.

求导并代入 x₁，计算出梯度 m = f'(x₁)。
Use the point-slope form: y – y₁ = m (x – x₁).

使用点斜式：y – y₁ = m (x – x₁)。

For the normal line, the gradient is the negative reciprocal: m_normal = -1/m (provided m ≠ 0). The normal is perpendicular to the tangent.

对于法线，其梯度为负倒数：m_normal = -1/m（前提是 m ≠ 0）。法线与切线垂直。

Example: Find the tangent to y = x² at x = 1. Derivative dy/dx = 2x, so at x=1 the gradient is 2. The point is (1,1). Tangent equation: y – 1 = 2(x – 1) -> y = 2x – 1.

例如：求 y = x² 在 x = 1 处的切线。导数为 dy/dx = 2x，所以在 x=1 处梯度为 2。点为 (1,1)。切线方程为 y – 1 = 2(x – 1)，即 y = 2x – 1。

6. Increasing and Decreasing Functions | 函数的单调性

A function is increasing on an interval if f'(x) > 0 for all x in that interval, and decreasing if f'(x) < 0. These conditions tell us where the graph slopes upward or downward.

若在某区间内对所有 x 都有 f'(x) > 0，则该函数在该区间递增；若 f'(x) < 0，则递减。这些条件表明了图像的上升或下降趋势。

To find the intervals of increase or decrease, solve the inequality f'(x) > 0 or f'(x) < 0. Use a sign diagram if necessary.

要求出递增或递减区间，需解不等式 f'(x) > 0 或 f'(x) < 0。必要时可使用符号表。

Example: For f(x) = x³ – 3x, f'(x) = 3x² – 3 = 3(x² – 1). f'(x) > 0 when x > 1 or x < -1, so the function increases on these intervals.

例如：对于 f(x) = x³ – 3x，f'(x) = 3x² – 3 = 3(x² – 1)。当 x > 1 或 x < -1 时 f'(x) > 0，因此函数在这些区间上递增。

7. Finding Stationary Points | 求驻点

Stationary points occur where the gradient is zero: dy/dx = 0. Solve this equation to find the x-coordinates of possible turning points.

驻点出现在梯度为零之处，即 dy/dx = 0。解此方程求出可能存在的转折点的 x 坐标。

Substitute each x-value back into the original function y = f(x) to get the corresponding y-coordinate. These (x, y) points are where the tangent is horizontal.

将每个 x 值代回原函数 y = f(x)，求得对应的 y 坐标。这些 (x, y) 点即为切线水平之处。

For y = x² – 4x + 1, dy/dx = 2x – 4. Setting 2x – 4 = 0 gives x = 2. Substituting gives y = (2)² – 4(2) + 1 = -3. Stationary point at (2, -3).

对于 y = x² – 4x + 1，dy/dx = 2x – 4。令 2x – 4 = 0 得 x = 2。代入得 y = (2)² – 4(2) + 1 = -3。驻点为 (2, -3)。

8. Classifying Stationary Points | 驻点分类（极大、极小、拐点）

To determine whether a stationary point is a maximum, minimum, or a stationary point of inflection, we can use the first derivative test or the second derivative test.

要判断一个驻点是极大值、极小值还是驻点拐点，我们可以使用一阶导数检验或二阶导数检验。

First derivative test: Check the sign of dy/dx just to the left and just to the right of the stationary point. If dy/dx changes from positive to negative -> maximum; from negative to positive -> minimum; no sign change -> inflection.

一阶导数检验：检查驻点左侧和右侧附近 dy/dx 的符号。若由正变负 → 极大值；由负变正 → 极小值；符号不变 → 拐点。

Second derivative test: Find d²y/dx² and evaluate at the stationary point. If d²y/dx² > 0 -> minimum; if d²y/dx² < 0 -> maximum; if d²y/dx² = 0 -> the test is inconclusive, use the first derivative test.

二阶导数检验：求 d²y/dx² 并在驻点处求值。若 d²y/dx² > 0 → 极小值；若 d²y/dx² < 0 → 极大值；若 d²y/dx² = 0 → 检验失效，需用一阶导数检验。

9. The Second Derivative in Detail | 二阶导数详解

The second derivative, d²y/dx² or f”(x), is obtained by differentiating the first derivative. It describes the rate of change of the gradient, or the concavity of the graph.

二阶导数 d²y/dx² 或 f”(x) 是由对一阶导数再次求导得到的。它描述了梯度的变化率，即图像的凹向。

If f”(x) > 0 on an interval, the graph is curved upwards (convex) and any stationary point there is a minimum. If f”(x) < 0, the graph is curved downwards (concave) and any stationary point is a

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GCSE CIE Maths: Differentiation Revision Notes | GCSE CIE 数学：微分考点精讲