GCSE CCEA Maths: Differentiation - Key Points | GCSE CCEA 数学：微分考点精讲

📚 GCSE CCEA Maths: Differentiation – Key Points | GCSE CCEA 数学：微分考点精讲

In GCSE CCEA Maths, differentiation is a fundamental calculus topic that allows you to find the gradient of a curve and analyse how functions change. This revision guide covers all key points, from the power rule to stationary points, ensuring you are fully prepared for the exam. We will walk through definitions, rules, worked examples, and common exam-style applications, always using a clear step-by-step approach.

在 GCSE CCEA 数学中，微分是一个基础的微积分专题，它让你能够求出曲线的梯度并分析函数的变化。本复习指南涵盖所有关键知识点，从幂函数法则到驻点，确保你为考试做好充分准备。我们将逐一讲解定义、法则、例题以及常见的考试题型，始终采用清晰的逐步推导方式。

1. What is Differentiation? | 什么是微分？

Differentiation is the process of calculating the derivative, which measures the rate of change of a function. If y = f(x), the derivative is denoted as dy/dx or f'(x). It tells you how fast y is changing with respect to x at any given instant.

微分是计算导数的过程，它衡量函数的变化率。如果 y = f(x)，导数记作 dy/dx 或 f'(x)。它告诉你 y 相对于 x 在任何给定瞬间的变化有多快。

The derivative gives the gradient of the tangent to the curve at any point. For a straight line, the gradient is constant, but for a curve, it varies at different points. This is why we need differentiation instead of simple ‘rise over run’.

导数给出了曲线在任意一点的切线梯度。对于直线，梯度是常数，但对于曲线，在不同点梯度会变化。这就是为什么我们需要微分，而不是简单的“对边比邻边”。

At GCSE level, you will mostly differentiate polynomial functions, but the concept extends to any smooth curve. Understanding dy/dx as a gradient function is the key to unlocking topics like stationary points and kinematics.

在 GCSE 阶段，你主要对多项式函数求导，但这一概念适用于任何光滑曲线。理解 dy/dx 作为梯度函数，是破解驻点、运动学等专题的关键。

2. The Power Rule | 幂函数法则

The most important rule for differentiating polynomials is the power rule. To apply it, multiply by the power and then reduce the power by 1.

对多项式求导最重要的法则是幂函数法则。应用方法是：将系数乘以幂指数，然后将幂指数减 1。

If y = xⁿ, then dy/dx = n xⁿ⁻¹

如果 y = xⁿ，那么 dy/dx = n xⁿ⁻¹

For example, if y = x³, then dy/dx = 3x². If y = x⁵, then dy/dx = 5x⁴. This rule works for any real number n, including fractions and negatives, but at GCSE you’ll mainly see positive integer powers and occasionally simple negative or fractional indices.

例如，如果 y = x³，那么 dy/dx = 3x²。如果 y = x⁵，那么 dy/dx = 5x⁴。该法则适用于任何实数 n，包括分数和负数，但在 GCSE 中你主要会见到正整数次幂，偶尔也会有简单的负指数或分数指数。

Remember that x⁰ = 1, so the derivative of just x is 1, and the derivative of a constant is 0. Always apply the power rule to each term individually when dealing with polynomials.

记住 x⁰ = 1，所以单独 x 的导数是 1，常数的导数是 0。在处理多项式时，一定要将幂函数法则分别应用于每一项。

3. Differentiating Polynomials | 多项式的求导

A polynomial is a sum of terms such as a xⁿ, where a is a constant coefficient. To differentiate, apply the power rule term by term: multiply each coefficient by its power, then decrease the power by one. Constants disappear because they have no rate of change.

多项式是像 a xⁿ 这样的项的和，其中 a 是常数系数。求导时逐项应用幂函数法则：将每个系数乘以该项的幂指数，然后将幂指数减一。常数项消失，因为它们没有变化率。

Let’s differentiate y = 4x³ + 2x² – 5x + 9. The derivative is dy/dx = 4·3x² + 2·2x¹ – 5·1x⁰ + 0 = 12x² + 4x – 5. Notice how the constant 9 simply becomes 0.

我们来求 y = 4x³ + 2x² – 5x + 9 的导数。导数为 dy/dx = 4·3x² + 2·2x¹ – 5·1x⁰ + 0 = 12x² + 4x – 5。注意常数 9 直接变成了 0。

When terms are subtracted, the sign stays with the coefficient. So -5x becomes -5. Always expand brackets before differentiating if necessary, but you’ll often meet expressions that are already in expanded form.

当各项相减时，符号跟随系数。因此 -5x 变成 -5。如果必要时，求导前一定先展开括号，但你遇到的很多表达式已经是展开形式了。

Practice makes perfect: try differentiating y = 6x⁴ – 3x³ + x – 2. You should get dy/dx = 24x³ – 9x² + 1.

熟能生巧：尝试求 y = 6x⁴ – 3x³ + x – 2 的导数。你应该得到 dy/dx = 24x³ – 9x² + 1。

4. The Gradient of a Curve | 曲线的梯度

To find the gradient at a specific point on a curve, first differentiate to obtain dy/dx, which is the gradient function. Then substitute the x-coordinate of the point into this function. The result is the numeric gradient at that exact point.

要求曲线上某一点的梯度，首先求导得到梯度函数 dy/dx，然后将该点的 x 坐标代入这个函数。结果就是该点处的具体梯度值。

Gradient at point (a, f(a)) = f'(a)

点 (a, f(a)) 处的梯度 = f'(a)

For y = x² + 4x + 1, find the gradient at x = 3. dy/dx = 2x + 4. At x = 3, gradient = 2(3) + 4 = 10. This means the curve is rising steeply at that point.

对于 y = x² + 4x + 1，求 x = 3 处的梯度。dy/dx = 2x + 4。当 x = 3 时，梯度 = 2(3) + 4 = 10。这意味着曲线在该点正陡峭上升。

A positive gradient means the function is increasing as x increases; a negative gradient means it is decreasing; zero gradient indicates a stationary point. These ideas help you understand the shape of a graph without plotting every point.

正梯度表示随着 x 增大函数递增；负梯度表示函数递减；零梯度表示一个驻点。这些概念帮助你无需逐点描画即可理解图形的大致形状。

5. Finding the Equation of a Tangent | 求切线方程

The tangent is a straight line that touches the curve at exactly one point and shares the curve’s gradient there. To find its equation, you need the gradient at the point of contact and the coordinates of that point. Then use the point-slope formula.

切线是一条刚好接触曲线于一点、且在该点与曲线梯度相同的直线。要求其方程，你需要切点处的梯度和该点的坐标，然后使用点斜式公式。

y – y₁ = m (x – x₁)

Steps: (1) Differentiate to get the gradient function m(x). (2) Substitute the given x₁ to find the numeric gradient m. (3) Find y₁ by plugging x₁ into the original equation. (4) Substitute m, x₁, y₁ into y – y₁ = m(x – x₁) and rearrange into the required form, often y = mx + c.

步骤：(1) 求导得到梯度函数 m(x

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GCSE CCEA Maths: Differentiation – Key Points | GCSE CCEA 数学：微分 考点精讲