📚 Differentiation: Rules, Tangents and Rates of Change | 微分：法则、切线与变化率
Differentiation is a core tool in A-level mathematics. It measures how a function changes as its input changes, giving the slope of a tangent and enabling optimisation, curve sketching and kinematics problems.
微分是 A-level 数学的核心工具。它度量函数随自变量变化的快慢，给出切线的斜率，并用于最优化、曲线草图和运动学问题。
This revision guide covers the main derivative rules, tangent and normal equations, stationary points, concavity and real-world rates of change.
本复习指南涵盖主要求导法则、切线与法线方程、驻点、凹凸性以及现实中的变化率。
1. The Meaning of the Derivative | 导数的含义
tion finds the instantaneous rate of change of a function. Geometrically, the derivative f'(x) is the gradient of the tangent to the curve y = f(x) at a point.
微分是求函数瞬时变化率的方法。从几何上看，导数 f'(x) 是曲线 y = f(x) 在某一点处切线的斜率。
The formal limit definition is f'(x) = limh→0 [f(x+h) − f(x)] / h. In practice, we use standard rules rather than this limit every time.
形式化极限定义为 f'(x) = limh→0 [f(x+h) − f(x)] / h。实际计算中，我们通常直接使用标准法则，而不会每次都求极限。
Common notations include f'(x), dy/dx and y'. They all mean the derivative of y with respect to x.
常见记号有 f'(x)、dy/dx 和 y'。它们都表示 y 对 x 的导数。
2. The Power Rule | 幂函数法则
the derivative is dy/dx = nxⁿ⁻¹. Multiply by the old power, then reduce the power by one.
若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。先乘以原来的指数，再把指数减 1。
Example: y = x⁵ gives dy/dx = 5x⁴. For y = x⁻³, dy/dx = −3x⁻⁴. For y = √x = x1/2, dy/dx = ½ x−1/2.
例如：y = x⁵ 的导数为 dy/dx = 5x⁴；y = x⁻³ 的导数为 −3x⁻⁴；y = √x = x1/2 的导数为 ½ x−1/2。
The power rule is the foundation for most A-level differentiation questions.
幂函数法则是大多数 A-level 微分题的基础。
3. Sum, Difference and Constant Multiples | 和、差与常数倍法则
Differentiation is linear, so constants factor out and terms can be differentiated separately.
微分具有线性性质，因此常数因子可以提出，各加项可以分别求导。
4x³ − 7x² + 2x − 9, then dy/dx = 12x² − 14x + 2. The constant term −9 differentiates to 0.
若 y = 4x³ − 7x² + 2x − 9，则 dy/dx = 12x² − 14x + 2。常数项 −9 的导数为 0。
Always rewrite fractional and negative powers before differentiating. For example, 1/x² becomes x⁻².
在求导前，应先把分式和负指数形式改写清楚。例如，1/x² 可写为 x⁻²。
4. The Product Rule | 乘积法则
When y = u(x)v(x), use dy/dx = u dv/dx + v du/dx. This is the product rule.
当 y = u(x)v(x) 时，使用 dy/dx = u dv/dx + v du/dx。这就是乘积法则。
Example: y = x² sin x. Let u = x² and v = sin x, so u' = 2x and v' = cos x. Hence dy/dx = x² cos x + 2x sin x.
例如：y = x² sin x。令 u = x²，v = sin x，则 u' = 2x，v' = cos x。因此 dy/dx = x² cos x + 2x sin x。
Always write the two parts clearly before differentiating to avoid missing a term in the final answer.
求导前先把两个部分写清楚，可以避免最终答案中漏项。
5. The Quotient Rule | 商法则
is dy/dx = (v du/dx − u dv/dx) / v². The denominator is the square of the bottom function.
若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。分母是原下方函数的平方。
2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1)/(x − 2)² = (x² − 4x − 1)/(x − 2)².
例如：y = (x² + 1)/(x − 2)。令 u = x² + 1，v = x − 2，得到 − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1)/(x − 2)² = (x² − 4x − 1)/(x − 2)²。
A common mistake is reversing the numerator as u dv/dx − v du/dx. For quotients, the bottom times derivative of top comes first.
常见错误是把分子写成 u dv/dx − v du/dx。商法则中，必须先写下方函数乘以上方函数的导数。
6. The Chain Rule | 链式法则
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