📚 A-Level Edexcel Pure Maths: Differentiation Rules and Applications | A-Level Edexcel 纯数学：微分法则与应用
Differentiation is one of the most important topics in Edexcel A-Level Pure Mathematics. It allows us to measure how fast a function is changing at any point, and it is essential for problems involving tangents, stationary points, curve sketching, and optimisation. This article covers the key rules, standard derivatives, and common exam applications you need to master.
微分是 Edexcel A-Level 纯数学中最重要的主题之一。它使我们能够度量函数在任意点的变化速度，对于涉及切线、驻点、曲线作图和优化的问题至关重要。本文涵盖你需要掌握的关键法则、标准导数以及常见考试应用。
1. Introduction to Differentiation | 微分入门
Differentiation measures the instantaneous rate of change of a function. For a curve y = f(x), the derivative f'(x) represents the gradient of the tangent at any point (x, y).
微分用于度量函数的瞬时变化率。对于曲线 y = f(x)，导数 f'(x) 表示任意点 (x, y) 处切线的斜率。
The notation dy/dx or f'(x) is used throughout Edexcel A-Level Pure Mathematics. Being comfortable with both forms is essential, as questions may switch between them.
在 Edexcel A-Level 纯数学中，全程使用记号 dy/dx 或 f'(x)。熟练运用这两种形式非常重要，因为题目可能会在它们之间转换。
For example, if y = x², then dy/dx = 2x. At x = 3, the gradient of the tangent is 6, meaning the curve rises steeply at that point.
例如，若 y = x²，则 dy/dx = 2x。在 x = 3 处，切线斜率为 6，这意味着曲线在该点上升较陡。
2. Differentiation from First Principles | 从第一原理求导
The formal definition of the derivative is the limit of the average rate of change as the interval h tends to zero.
导数的正式定义是当间隔 h 趋于零时平均变化率的极限。
f'(x) = lim (h → 0) [f(x+h) – f(x)] / h
You must be able to apply this definition to simple functions such as x², x³, and 1/x. Expand and simplify the difference quotient before taking the limit.
你必须能够将这个定义应用于简单函数，如 x²、x³ 和 1/x。在取极限之前，先展开并化简差商。
Example: For f(x) = x², the difference quotient becomes (2xh + h²)/h = 2x + h. As h → 0, the limit is 2x, confirming the power rule.
例子：对于 f(x) = x²，差商变为 (2xh + h²)/h = 2x + h。当 h → 0 时，极限为 2x，从而验证了幂法则。
3. Basic Differentiation Rules | 基本微分法则
The power rule states that if y = xⁿ, then dy/dx = n xⁿ⁻¹, for any real power n. This is the most frequently used rule in A-Level differentiation.
幂法则指出，若 y = xⁿ，则 dy/dx = n xⁿ⁻¹，适用于任意实数幂 n。这是 A-Level 微分中最常用的法则。
d/dx (xⁿ) = n xⁿ⁻¹
The constant multiple rule and sum/difference rule allow you to differentiate term by term. Constants disappear because their rate of change is zero.
常数倍法则与和差法则允许你逐项求导。常数项消失是因为它们的变化率为零。
Example: If y = 3x⁴ – 5x² + 2x – 7, then dy/dx = 12x³ – 10x + 2. Notice that the constant –7 becomes 0.
例子：若 y = 3x⁴ – 5x² + 2x – 7，则 dy/dx = 12x³ – 10x + 2。注意常数 –7 变为 0。
4. Chain Rule | 链式法则
The chain rule is used for composite functions y = f(g(x)). It states that dy/dx = f'(g(x)) × g'(x), or in Leibniz notation:
链式法则用于复合函数 y = f(g(x))。它指出 dy/dx = f'(g(x)) × g'(x)，或用莱布尼茨记号表示为：
dy/dx = (dy/du) × (du/dx)
Write the inner function as u, differentiate both y with respect to u and u with respect to x, then multiply the two results.
将内层函数设为 u，分别对 y 关于 u 和 u 关于 x 求导，然后相乘得到结果。
Example: y = (2x + 1)⁵, let u = 2x + 1, then dy/dx = 5u⁴ × 2 = 10(2x + 1)⁴. The chain rule is especially useful for powers of linear functions.
例子：y = (2x + 1)⁵，设 u = 2x + 1，则 dy/dx = 5u⁴ × 2 = 10(2x + 1)⁴。链式法则对于线性函数的幂特别有用。
5. Product Rule | 乘积法则
When y = u v, where u and v are both functions of x, the derivative is given by dy/dx = u’ v + u v’.
当 y = u v，其中 u 和 v 都是 x 的函数时，导数为 dy/dx = u’ v + u v’。
d/dx (u v) = u’ v + u v’
You must be careful to keep the original functions intact in each term. A common mistake is to write only the derivatives, but the product rule requires one differentiated factor and one original factor in each term.
你必须注意在每一项中保持原始函数不变。一个常见错误是只写导数，但乘积法则要求每一项中一个是导数因子，另一个是原始因子。
Example: y = x² sin x, then dy/dx = 2x sin x + x² cos x. Here u = x² and v = sin x, so u’ = 2x and v’ = cos x.
例子：y = x² sin x，则 dy/dx = 2x sin x + x² cos x。这里 u = x²，v = sin x，因此 u’ = 2x，v’ = cos x。
6. Quotient Rule | 商法则
For y = u / v, where u and v are functions of x, the derivative is dy/dx = (u’ v – u v’) / v².
对于 y = u / v，其中 u 和 v 是 x 的函数，导数为 dy/dx = (u’ v – u v’) / v²。
d/dx (u / v) = (u’ v – u v’) / v²
The order in the numerator matters: it is u’v minus uv’, not the reverse. Getting this sign wrong is a very common error in exams.
分子中的顺序很重要：是 u’v 减去 uv’，不可颠倒。在考试中写错这个符号是一个非常常见的错误。
Example: y = x / (x² + 1), then dy/dx = [(1)(x²+1) – x(2x)] / (x²+1)² = (1 – x²) / (x²+1)². Always simplify the numerator if possible.
例子：y = x / (x² + 1)，则 dy/dx = [(1)(x²+1) – x(2x)] / (x²+1)² = (1 – x²) / (x²+1)²。如果可能，始终化简分子。
7. Differentiating Standard Functions | 标准函数求导
You need to memorise the derivatives of eˣ, ln x, sin x, cos x, and tan x. These appear constantly in Edexcel exam questions.
你需要熟记 eˣ、ln x、sin x、cos x 和 tan x 的导数。它们经常出现在 Edexcel 考试题中。
| Function | Derivative |
|---|---|
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| tan x | sec² x |
These derivatives often appear in combination with chain, product, or quotient rules. For instance, the derivative of e^(2x) is 2e^(2x), using the chain rule.
这些导数经常与链式法则、乘积法则或商法则结合出现。例如，e^(2x) 的导数是 2e^(2x)，使用了链式法则。
Also be aware that the derivative of ln(ax + b) is a/(ax + b), and the derivative of sin(ax + b) is a cos(ax + b).
还要注意 ln(ax + b) 的导数是 a/(ax + b)，sin(ax + b) 的导数是 a cos(ax + b)。
8. Higher Derivatives | 高阶导数
The second derivative f”(x) or d²y/dx² measures the rate of change of the gradient. It is found by differentiating the first derivative.
二阶导数 f”(x) 或 d²y/dx² 度量斜率的变化率。它通过对一阶导数再次求导得到。
d²y/dx² = d/dx (dy/dx)
It is used to classify stationary points as maxima, minima, or points of inflection. If the second derivative is positive, the curve is concave up; if negative, concave down.
它用于将驻点分类为极大值、极小值或拐点。如果二阶导数为正，曲线向上凹；如果为负，曲线向下凹。
You may also be asked to find third or higher derivatives in Edexcel exams, especially in questions on Maclaurin series or repeated differentiation.
在 Edexcel 考试中，你也可能被要求求三阶或更高阶导数，尤其是在麦克劳林级数或重复求导的问题中。
9. Tangents and Normals | 切线与法线
The gradient of the tangent at x = a is m = f'(a). The equation of the tangent is y – y₁ = m(x – x₁), where (x₁, y₁) is the point on the curve.
在 x = a 处切线的斜率是 m = f'(a)。切线方程为 y – y₁ = m(x – x₁)，其中 (x₁, y₁) 是曲线上的点。
The normal is perpendicular to the tangent, so its gradient is –1/m, provided m ≠ 0. If the tangent is horizontal, the normal is vertical.
法线垂直于切线，所以其斜率为 –1/m，前提是 m ≠ 0。如果切线是水平的，法线就是竖直的。
Example: For y = x² at the point (3, 9), m = 6. The tangent equation is y – 9 = 6(x – 3), and the normal gradient is –1/6.
例子：对于 y = x² 在点 (3, 9) 处，m = 6。切线方程为 y – 9 = 6(x – 3)，法线斜率为 –1/6。
10. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where dy/dx = 0. To classify them, find the second derivative or test the sign of dy/dx on either side of the point.
驻点出现在 dy/dx = 0 处。为了对其分类，求二阶导数或检验该点两侧 dy/d
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