📚 Edexcel A-Level Pure Maths: Differentiation Rules and Applications | 爱德思A-Level纯数学：微分法则与应用
Differentiation is one of the most heavily examined topics in the Edexcel A-Level Pure Mathematics specification. It develops the idea of a gradient from straight lines to curves, and it provides the mathematical machinery for optimisation, curve sketching, kinematics and many applied problems.
微分是爱德思A-Level纯数学大纲中考查最频繁的主题之一。它将直线的梯度概念扩展到曲线，并为最优化、曲线作图、运动学以及许多应用问题提供了数学工具。
1. The Limit Definition | 极限定义
Differentiation from first principles defines the derivative of a function f at a point x as the limit of the gradient of a chord joining (x, f(x)) and (x+h, f(x+h)) as h tends to zero. This formal definition is important because it links the derivative to the idea of instantaneous rate of change.
从第一性原理求导将函数 f 在点 x 处的导数定义为连接 (x, f(x)) 与 (x+h, f(x+h)) 的割线斜率在 h 趋于零时的极限。这个正式定义之所以重要，是因为它将导数与瞬时变化率的概念联系起来。
f'(x) = lim h→0 [f(x+h) − f(x)] / h
For example, if f(x) = x², the expansion gives f(x+h) − f(x) = 2xh + h². Dividing by h and letting h → 0 leaves the derivative f'(x) = 2x.
例如，若 f(x) = x²，展开后得到 f(x+h) − f(x) = 2xh + h²。除以 h 并令 h → 0，就得到导数 f'(x) = 2x。
- English: f'(x) is read as f prime of x. 中文：f'(x) 读作 f prime of x。
- English: The notation dy/dx means the derivative of y with respect to x. 中文：dy/dx 表示 y 关于 x 的导数。
2. Power Rule and Sum Rule | 幂函数法则与加减法则
The power rule states that if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹, where n is any real constant. This rule is the foundation for differentiating polynomials and many rational or radical functions.
幂函数法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 是任意实数常数。这个法则是求多项式以及许多有理函数或根式函数导数的基础。
d/dx (xⁿ) = n xⁿ⁻¹
Constants and sums are handled separately: the derivative of a constant is zero, and the derivative of a sum is the sum of the derivatives. Therefore d/dx [5x³ − 2x + 7] = 15x² − 2.
常数与和式可以分别处理：常数的导数为零，和式的导数等于各项导数的和。因此 d/dx [5x³ − 2x + 7] = 15x² − 2。
- English: Differentiate term by term. 中文：逐项求导。
- English: Rewrite √x as x¹ᐟ² before using the power rule. 中文：在使用幂函数法则前，先将 √x 改写为 x¹ᐟ²。
3. Chain Rule | 链式法则
The chain rule is used when one function is composed with another. If y = f(u) and u = g(x), then dy/dx is the product of dy/du and du/dx.
链式法则用于一个函数与另一个函数复合的情况。若 y = f(u) 且 u = g(x)，则 dy/dx 等于 dy/du 与 du/dx 的乘积。
dy/dx = dy/du × du/dx
For example, let y = (3x² + 1)⁵. Set u = 3x² + 1, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x. Multiplying gives dy/dx = 30x(3x² + 1)⁴.
例如，设 y = (3x² + 1)⁵。令 u = 3x² + 1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x。相乘得到 dy/dx = 30x(3x² + 1)⁴。
- English: Identify the inner function u clearly. 中文：清楚识别内层函数 u。
- English: Always multiply by du/dx, not just by u’. 中文：始终乘以 du/dx，而不是只乘以 u’。
4. Product Rule | 乘法法则
When differentiating a product of two functions y = u(x)v(x), Edexcel expects you to quote and apply the product rule accurately. The derivative is u times the derivative of v plus v times the derivative of u.
当对两个函数的乘积 y = u(x)v(x) 求导时，爱德思考试要求准确引用并应用乘法法则。导数等于 u 乘以 v 的导数再加上 v 乘以 u 的导数。
d/dx (uv) = u dv/dx + v du/dx
For y = x² sin x, take u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x, so dy/dx = 2x sin x + x² cos x.
对于 y = x² sin x，取 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，所以 dy/dx = 2x sin x + x² cos x。
- English: State u and v before substituting. 中文：在代入前先写出 u 和 v。
- English: The order of terms does not matter, but signs must be kept exactly. 中文：项的顺序无关紧要，但符号必须完全保留。
5. Quotient Rule | 除法法则
For a quotient y = u/v, the derivative is found using the denominator squared in the denominator of the result. The numerator is v du/dx minus u dv/dx.
对于商式 y = u/v，其导数的分母为原分母的平方，分子为 v du/dx 减去 u dv/dx。
d/dx (u/v) = (v du/dx − u dv/dx) / v²
If y = x / (x² + 1), then u = x, v = x² + 1, du/dx = 1 and dv/dx = 2x. The derivative is (x² + 1) − x(2x) all over (x² + 1)², which simplifies to (1 − x²) / (x² + 1)².
若 y = x / (x² + 1)，则 u = x，v = x² + 1，du/dx = 1，dv/dx = 2x。导数为 [(x² + 1) − x(2x)] / (x² + 1)²，化简为 (1 − x²) / (x² + 1)²。
- English: Write the denominator as v², not as v times itself in a cancel form. 中文：把分母写成 v²，不要写成可以约分的 v 乘以自身。
- English: Be careful with the minus sign in the numerator. 中文：注意分子中的负号。
6. Exponentials and Logarithms | 指数函数与对数函数求导
Exponential and logarithmic functions have distinctive derivatives. The exponential function eˣ is its own derivative, while the natural logarithm ln x differentiates to the reciprocal of x.
指数函数与对数函数有独特的导数。指数函数 eˣ 的导数就是它本身，而自然对数 ln x 的导数为 x 的倒数。
d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x
For a composite exponential such as y = eᵏˣ, the chain rule gives dy/dx = k eᵏˣ. Similarly, d/dx [ln(kx)] = 1/x because the constant k cancels after differentiation.
对于像 y = eᵏˣ 这样的复合指数函数，链式法则给出 dy/dx = k eᵏˣ。类似地，d/dx [ln(kx)] = 1/x，因为常数 k 在求导后会被消去。
| Function f(x) | Derivative f'(x) |
|---|---|
| eˣ | eˣ |
| eᵏˣ | k eᵏˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
7. Trigonometric Differentiation | 三角函数求导
The standard trigonometric derivatives must be memorised accurately. Edexcel often tests these results in combination with the product, quotient and chain rules.
标准的三角函数导数必须准确记忆。爱德思考试经常将这些结果与乘法法则、除法法则和链式法则结合起来考查。
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
For example, y = sin² x can be treated as y = (sin x)². Using the chain rule gives dy/dx = 2 sin x cos x, which is the same as sin 2x.
例如，y = sin² x 可以看作 y = (sin x)²。使用链式法则得到 dy/dx = 2 sin x cos x，这也等于 sin 2x。
- English: Note the negative sign in the derivative of cos x. 中文：注意 cos x 导数中的负号。
- English: Express tan x as sin x / cos x to derive sec² x if required. 中文：若需要，可将 tan x 写为 sin x / cos x 来推导 sec² x。
8. Implicit Differentiation | 隐函数求导
When a relationship between x and y is given implicitly, differentiate both sides with respect to x, treating y as a function of x. Every derivative of a y-term must include a factor dy/dx.
当 x 与 y 的关系以隐函数形式给出时，对等式两边关于 x 求导，并将 y 视为 x 的函数。每个含 y 的项求导后都必须乘上因子 dy/dx。
d/dx (y²) = 2y dy/dx
For the circle x² + y² = 25, differentiating both sides gives 2x + 2y dy/dx = 0. Solving gives dy/dx = −x / y.
对于圆 x² + y² = 25，对两边求导得到 2x + 2y dy/dx = 0。解得 dy/dx = −x / y。
- English: Apply the product rule to mixed terms such as xy. 中文：对 xy 这样的混合项使用乘法法则。
- English: Do not forget the dy/dx factor on every y derivative. 中文：不要忘记每个 y 的导数都要带有 dy/d
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