📚 Differentiation: First Principles, Rules and Applications | 微分：第一原理、求导法则与应用
Differentiation is one of the two central ideas in A-Level calculus. It measures the instantaneous rate of change of a function. In Edexcel Pure Mathematics, you are expected to define the derivative from first principles, apply standard rules accurately, and use differentiation to solve problems involving tangents, normals, stationary points and modelling.
微分是 A-Level 微积分的两大核心思想之一，用来度量函数的瞬时变化率。在 Edexcel 纯数学考试中，你需要掌握从第一原理定义导数，熟练运用求导法则，并利用微分解决切线、法线、驻点和建模等问题。
1. Gradient of a Chord | 割线斜率
The gradient of the curve y = f(x) between x = a and x = a + h is given by [f(a + h) − f(a)] / h. This is the average rate of change over the small interval h.
曲线 y = f(x) 在 x = a 与 x = a + h 之间的割线斜率是 [f(a + h) − f(a)] / h，它是小区间 h 上的平均变化率。
As h gets smaller, the chord gets closer to the tangent at x = a. The limiting value of the chord gradient is therefore the gradient of the curve at that point.
当 h 越来越小时，割线越来越接近 x = a 处的切线。因此，割线斜率的极限值就是曲线在该点的切线斜率。
2. First Principles Definition | 第一原理定义
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
The derivative f'(x) is the limit of the difference quotient as h tends to zero. This limit measures the instantaneous rate of change of f(x) with respect to x.
导数 f'(x) 是差商的极限，其中 h 趋于 0。这个极限度量了 f(x) 关于 x 的瞬时变化率。
For example, if f(x) = x², then f'(x) = lim(h→0) [(x + h)² − x²] / h = lim(h→0) (2xh + h²) / h = lim(h→0) (2x + h) = 2x. This shows that the gradient of y = x² is exactly 2x.
例如，若 f(x) = x²，则 f'(x) = lim(h→0) [(x + h)² − x²] / h = lim(h→0) (2xh + h²) / h = lim(h→0) (2x + h) = 2x。这说明 y = x² 的斜率正是 2x。
3. Differentiating xⁿ | 幂函数求导
d/dx (xⁿ) = n xⁿ⁻¹
For any real constant n, the derivative of xⁿ is n xⁿ⁻¹. This is usually called the power rule and is the most frequently used rule in A-Level differentiation.
对于任意实常数 n，xⁿ 的导数为 n xⁿ⁻¹。这就是通常所说的幂法则，也是 A-Level 微分中最常用的法则。
Examples: d/dx (x⁵) = 5x⁴, d/dx (x⁻²) = −2x⁻³, and d/dx (x¹/²) = ½ x⁻¹/² = 1/(2√x).
示例：d/dx (x⁵) = 5x⁴，d/dx (x⁻²) = −2x⁻³，以及 d/dx (x¹/²) = ½ x⁻¹/² = 1/(2√x)。
4. Constant Multiples and Sums | 常数倍与和差法则
d/dx [c f(x)] = c f'(x)
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Constants can be moved outside the derivative, and differentiation can be carried out term by term. These results allow you to differentiate polynomials and other linear combinations quickly.
常数可以提到导数符号外，求导可以逐项进行。这些结论使你能快速对多项式以及其他线性组合求导。
For example, d/dx (3x⁴ − 5x² + 7x − 2) = 12x³ − 10x + 7. The constant term −2 differentiates to 0.
例如，d/dx (3x⁴ − 5x² + 7x − 2) = 12x³ − 10x + 7。常数项 −2 的导数为 0。
5. Chain Rule | 链式法则
dy/dx = dy/du × du/dx
When y is a function of u and u is a function of x, the overall rate of change is the product of the two separate rates of change. The chain rule is essential for composite functions such as (ax + b)ⁿ, eᵏˣ, ln(g(x)) and sin(g(x)).
当 y 是 u 的函数、u 是 x 的函数时，整体变化率等于两个变化率的乘积。链式法则对于 (ax + b)ⁿ、eᵏˣ、ln(g(x)) 和 sin(g(x)) 等复合函数至关重要。
An equivalent form is d/dx [f(g(x))] = f'(g(x)) g'(x). For example, if y = (3x² + 5)⁴, let u = 3x² + 5, then dy/dx = 4u³ × 6x = 24x(3x² + 5)³.
等价形式为 d/dx [f(g(x))] = f'(g(x)) g'(x)。例如，若 y = (3x² + 5)⁴，令 u = 3x² + 5，则 dy/dx = 4u³ × 6x = 24x(3x² + 5)³。
6. Product Rule | 乘积法则
d/dx [u v] = u dv/dx + v du/dx
If y = u(x)v(x), differentiate u and v separately and combine them as shown. The product rule must be used whenever two non-constant functions are multiplied.
如果 y = u(x)v(x)，分别求出 u 和 v 的导数，再按上述公式组合。只要两个非常数函数相乘，就必须使用乘积法则。
Example: if y = x²eˣ, then dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2). The factor eˣ is often left in the final answer.
示例：若 y = x²eˣ，则 dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2)。因子 eˣ 通常会保留在最终答案中。
7. Quotient Rule | 商法则
d/dx [u/v] = (v du/dx − u dv/dx) / v²
For y = u/v, the derivative denominator is v². The order in the numerator is crucial: it is v du/dx minus u dv/dx, not the reverse.
对于 y = u/v，导数分母为 v²。分子中的顺序非常关键：是 v du/dx 减 u dv/dx，而不是反过来。
Example: y = (x² + 1)/(x − 2), so dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1)/(x − 2)² = (x² − 4x − 1)/(x − 2)².
示例：y = (x² + 1)/(x − 2)，所以 dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (2x² − 4x − x² − 1)/(x − 2)² = (x² − 4x − 1)/(x − 2)²。
8. Standard Derivatives | 常见函数导数
These standard results must be memorised because they appear regularly in chain, product and quotient rule questions. They form the building blocks of more complex differentiation.
以下标准结果必须记住，因为它们经常出现在链式法则、乘积法则和商法则的题目中。它们是更复杂求导的基础构件。
| f(x) | f'(x) |
|---|---|
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| cot x | −cosec² x |
9. Second Derivative and Stationary Points | 二阶导数与驻点
The second derivative f”(x) measures the rate of change of f'(x). At a stationary point, f'(x) = 0, meaning the tangent to the curve is horizontal.
二阶导数 f”(x) 度量 f'(x) 的变化率。在驻点处，f'(x) = 0，这意味着曲线的切线是水平的。
If f”(a) > 0, the stationary point at x = a is a local minimum. If f”(a) < 0, it is a local maximum. If f''(a) = 0, the second derivative test is inconclusive and you should check the sign of f'(x) on either side of x = a.
如果 f”(a) > 0，则 x = a 处的驻点是局部极小值。如果 f”(a) < 0，则为局部极大值。如果 f''(a) = 0，二阶导数检验无法得出结论，此时应检查 f'(x) 在 x = a 两侧的符号。
Stationary points are also called turning points in Edexcel exam questions, and you may be asked to determine their nature using the second derivative or a sign table.
在 Edexcel 考试题中，驻点也称为转折点，你可能会被要求使用二阶导数或符号表来判断其性质。
10. Tangents and Normals | 切线与法线
At x = a, the tangent has gradient m = f'(a). Its equation is given by y − f(a) = m(x − a).
在 x = a 处，切线的斜率为 m = f'(a)，其方程为 y − f(a) = m(x − a)。
The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0. The normal has the same point of contact but a different slope.
法线垂直于切线，因此其斜率为 −1/m，前提是 m ≠ 0。法线经过同一个切点，但斜率不同。
Example: for y = x³ − 3x at x = 1, f'(x) = 3x² − 3, so m = 0 and f(1) = −2. The tangent is y = −2; the normal is x = 1.
示例：对于 y = x³ − 3x，在 x = 1 处，f'(x) = 3x² − 3，所以 m = 0，f(1) = −2。切线为 y = −2；法线为 x = 1。
11. Rates of Change and Modelling | 变化率与建模
Differentiation allows you to connect related rates. If V is volume and r is radius, then dV/dt = dV/dr × dr/dt. This chain rule form is common in practical problems.
微分可以联系相关变化率。若 V 为体积、r 为半径，则 dV/dt = dV/dr × dr/dt。这种链式法则形式在实际问题中很常见。
In kinematics, displacement s, velocity v and acceleration a are linked by v = ds/dt and a = dv/dt = d²s/dt². Understanding the derivative as a rate of change is essential for interpreting models.
在运动学中，位移 s、速度 v 和加速度 a 满足 v = ds/dt 以及 a = dv/dt = d²s/dt²。将导数理解为变化率对于解释模型至关重要。
12. Common Exam Mistakes | 常见考试错误
- Forgetting the chain rule when differentiating compound functions — 求导复合函数时忘记链式法则。
- Confusing the order in the quotient rule numerator — 商法则分子中的顺序写反。
- Not simplifying f'(x) before substituting stationary points — 代入驻点前未化简 f'(x)。
- Treating aˣ as if it followed the xⁿ rule; in fact d/dx (aˣ) = aˣ ln a — 把 aˣ 当作 xⁿ 法则处理；实际上 d/dx (aˣ) = aˣ ln a。
- Forgetting the negative sign when differentiating cos x or cot x — 求导 cos x 或 cot x 时忘记负号。
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