📚 GCSE WJEC Mathematics: Differentiation Revision | GCSE WJEC 数学：微分 考点精讲
Differentiation is a cornerstone of GCSE Higher Tier Mathematics, enabling us to analyse how functions change. In WJEC exams, you’ll be expected to differentiate polynomial functions, find gradients of curves, determine equations of tangents, and apply differentiation to solve real-world problems involving maximum and minimum values. This revision guide covers every key concept and technique you need, with clear examples and exam-focused tips.
微分是 GCSE 高等级数学的基石，帮助我们分析函数的变化。在 WJEC 考试中，你需要掌握多项式函数的求导、求曲线梯度、确定切线方程，并应用微分解决涉及最大值和最小值的实际问题。本复习指南涵盖所有关键概念和技巧，配有清晰的示例和考试要点提示。
1. What is Differentiation? | 什么是微分？
Differentiation is the process of finding the derivative of a function, which measures the instantaneous rate of change. Geometrically, the derivative at a point equals the gradient of the tangent to the curve at that point. For a function y = f(x), the derivative is denoted dy/dx or f'(x).
微分是求函数导数的一个过程，导数衡量的是瞬时变化率。从几何角度看，某一点的导数等于该点处曲线切线的斜率。对于函数 y = f(x)，导数记作 dy/dx 或 f'(x)。
Unlike straight-line graphs which have constant gradient, curves have varying steepness. Differentiation allows us to find the gradient at any specific point on a curve without drawing a tangent by hand.
直线图具有恒定斜率，而曲线的陡峭程度各处不同。利用微分，我们无需手动画切线即可求出曲线上任意指定点的梯度。
2. The Power Rule: Differentiating xⁿ | 幂法则：对 xⁿ 求导
The most fundamental rule is the power rule: if y = xⁿ, then dy/dx = n xⁿ⁻¹. This means you multiply by the power, then reduce the power by one.
最基本的是幂法则：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。这意味着先将指数乘下来作为系数，然后将指数减 1。
For example, differentiate y = x⁵ → dy/dx = 5x⁴; differentiate y = x⁻² → dy/dx = -2x⁻³; and for y = √x = x¹/², dy/dx = (1/2)x⁻¹/² = 1/(2√x).
例如，对 y = x⁵ 求导得 dy/dx = 5x⁴；对 y = x⁻² 求导得 dy/dx = -2x⁻³；对于 y = √x = x¹/²，求导得 dy/dx = (1/2)x⁻¹/² = 1/(2√x)。
Remember that constants must be rewritten with a power of 0: y = 5 can be written as 5x⁰, differentiating gives 0. This aligns with the rule that the derivative of any constant is zero.
要记住常数可以写作指数为 0 的形式：y = 5 可写成 5x⁰，求导后得 0。这与常数的导数恒为零的规则一致。
| Function f(x) | Derivative f'(x) |
|---|---|
| x³ | 3x² |
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