A-Level Mathematics: Differentiation and Basic Rules of Differentiation | A-Level 数学：微分法与基本求导法则

📚 A-Level Mathematics: Differentiation and Basic Rules of Differentiation | A-Level 数学：微分法与基本求导法则

Differentiation is one of the most powerful tools in A-Level Mathematics. It allows us to calculate the rate at which one quantity changes with respect to another, and it forms the foundation of calculus. In this revision article, we will cover the definition of the derivative, the basic rules of differentiation, and how to apply them confidently in CIE A-Level exams.

微分法是 A-Level 数学中最强大的工具之一。它使我们能够计算一个量相对于另一个量的变化速率，是微积分的基础。在这篇复习文章中，我们将介绍导数的定义、基本求导法则以及如何在 CIE A-Level 考试中自信地运用它们。

1. The Definition of the Derivative | 导数的定义

The derivative of a function f(x) with respect to x is defined as the limit of the average rate of change as the interval approaches zero. Mathematically, we write:

f ′(x) = lim(h→0) [f(x + h) − f(x)] / h

This limit, when it exists, gives the slope of the tangent line to the curve y = f(x) at the point (x, f(x)). The notation f ′(x) is due to Lagrange, while dy/dx is Leibniz notation. Both are used in CIE examinations.

函数 f(x) 关于 x 的导数定义为当区间趋近于零时平均变化率的极限。数学上写作：

f ′(x) = lim(h→0) [f(x + h) − f(x)] / h

当该极限存在时，它给出曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。f ′(x) 是拉格朗日记法，dy/dx 是莱布尼茨记法。两者在 CIE 考试中都会出现。

2. Differentiability and Continuity | 可导性与连续性

For a function to be differentiable at a point, it must first be continuous there. However, continuity alone does not guarantee differentiability. A classic example is f(x) = |x| at x = 0, which is continuous but has a sharp corner, so it is not differentiable.

一个函数在某点可导，首先必须在该点连续。然而，仅连续并不能保证可导。经典例子是 f(x) = |x| 在 x = 0 处，该函数连续但有一个尖角，因此不可导。

Candidates should remember the following rules:

If f is differentiable at x = a, then f is continuous at x = a.
The converse is not true: a continuous function may fail to be differentiable at a point where it has a corner, a vertical tangent, or a discontinuity.

考生应记住以下规则：

若 f 在 x = a 处可导，则 f 在 x = a 处连续。
反之不成立：连续函数可能在尖点、垂直切线或间断点处不可导。

3. Basic Rules of Differentiation | 基本求导法则

The most fundamental rules are the constant rule, the power rule, and the constant multiple rule. These rules must be memorised perfectly because they apply to every differentiation problem.

最基本的法则是常数法则、幂法则和常数倍法则。这些必须完全记住，因为它们适用于每一个求导问题。

d/dx (c) = 0

d/dx (xⁿ) = n xⁿ⁻¹

d/dx (c f(x)) = c f ′(x)

For example, the derivative of 3x⁵ is 3 × 5x⁴ = 15x⁴. The derivative of 7 is 0 because a constant never changes.

例如，3x⁵ 的导数是 3 × 5x⁴ = 15x⁴。7 的导数是 0，因为常数永远不会变化。

4. The Sum and Difference Rules | 和差法则

When differentiating a sum or difference of terms, we differentiate each term separately and then add or subtract the results. This follows from the linearity of differentiation.

当对多项式或若干项的和差求导时，我们逐项求导然后将结果相加或相减。这源于微分运算的线性性质。

d/dx [f(x) ± g(x)] = f ′(x) ± g ′(x)

Consider y = 2x³ − 5x² + 3x − 8. Then dy/dx = 6x² − 10x + 3. Notice that the constant term −8 has derivative 0.

考虑 y = 2x³ − 5x² + 3x − 8，则 dy/dx = 6x² − 10x + 3。注意常数项 −8 的导数为 0。

5. The Product Rule | 乘积法则

When two functions are multiplied together, we cannot differentiate each factor individually. The product rule states that:

d/dx [u v] = u dv/dx + v du/dx

For example, let y = x² sin x. Here u = x² and v = sin x. Then dy/dx = x² cos x + 2x sin x. A common memory aid is “first times derivative of second plus second times derivative of first.”

当两个函数相乘时，我们不能各自单独求导。乘积法则告诉我们：

d/dx [u v] = u dv/dx + v du/dx

例如，设 y = x² sin x。这里 u = x²，v = sin x。则 dy/dx = x² cos x + 2x sin x。常用记忆口诀是“前导后不导加后导前不导”。

6. The Quotient Rule | 商法则

When dividing two functions, we use the quotient rule. It is essential to remember the correct order of terms to avoid sign errors.

当两个函数相除时，我们使用商法则。记清楚项的先后顺序以避免符号错误非常重要。

d/dx (u/v) = [v du/dx − u dv/dx] / v²

For example, let y = x / (1 + x²). Then dy/dx = [(1 + x²)(1) − x(2x)] / (1 + x²)² = (1 − x²) / (1 + x²)².

例如，设 y = x / (1 + x²)，则 dy/dx = [(1 + x²)(1) − x(2x)] / (1 + x²)² = (1 − x²) / (1 + x²)²。

7. The Chain Rule | 链式法则

The chain rule is used for composite functions, such as (3x + 1)⁷ or sin(2x). It states that:

dy/dx = (dy/du) × (du/dx)

Equivalently, if y = f(g(x)), then dy/dx = f′(g(x)) g′(x). For example, if y = (2x² + 3)⁵, let u = 2x² + 3. Then dy/du = 5u⁴ and du/dx = 4x, so dy/dx = 20x(2x² + 3)⁴.

链式法则用于复合函数，例如 (3x + 1)⁷ 或 sin(2x)。它表明：

dy/dx = (dy/du) × (du/dx)

等价地，若 y = f(g(x))，则 dy/dx = f′(g(x)) g′(x)。例如，若 y = (2x² + 3)⁵，令 u = 2x² + 3，则 dy/du = 5u⁴，du/dx = 4x，所以 dy/dx = 20x(2x² + 3)⁴。

8. Derivatives of Trigonometric Functions | 三角函数的导数

The derivatives of the six trigonometric functions are essential. In CIE A-Level, angles are measured in radians unless stated otherwise.

六个三角函数的导数是必需的。在 CIE A-Level 中，除非特别说明，角度均以弧度为单位。

f(x)	f ′(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

For example, d/dx [sin(3x)] = 3 cos(3x) by the chain rule. Remember the negative signs for cos, cosec and cot.

例如，d/dx [sin(3x)] = 3 cos(3x)，这是由链式法则得到的。注意 cos、cosec 和 cot 的负号。

9. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数

The exponential function eˣ has the remarkable property that its derivative is itself. The natural logarithm ln x has derivative 1/x. These results are indispensable.

指数函数 eˣ 有一个卓越的性质：它的导数是它自身。自然对数 ln x 的导数是 1/x。这些结果不可或缺。

d/dx (eˣ) = eˣ

d/dx (ln x) = 1/x

For a general base a, d/dx (aˣ) = aˣ ln a, and d/dx (logₐ x) = 1/(x ln a). For example, d/dx (2ˣ) = 2ˣ ln 2.

对于一般底数 a，d/dx (aˣ) = aˣ ln a，d/dx (logₐ x) = 1/(x ln a)。例如，d/dx (2ˣ) = 2ˣ ln 2。

10. Higher-Order Derivatives | 高阶导数

If we differentiate a function twice, we obtain the second derivative, written as f ″(x) or d²y/dx². Higher-order derivatives are used to determine concavity and points of inflection, and appear in kinematics as acceleration.

如果对函数求导两次，就得到二阶导数，记作 f ″(x) 或 d²y/dx²。高阶导数用于判断凹凸性和拐点，在运动学中表现为加速度。

For example, if y = x⁴ − 3x² + 5, then dy/dx = 4x³ − 6x, and d²y/dx² = 12x² − 6. In kinematics, if displacement s is given as a function of time t, then ds/dt is velocity and d²s/dt² is acceleration.

例如，若 y = x⁴ − 3x² + 5，则 dy/dx = 4x³ − 6x，d²y/dx² = 12x² − 6。在运动学中，若位移 s 是时间 t 的函数，则 ds/dt 是速度，d²s/dt² 是加速度。

11. Implicit Differentiation | 隐函数求导

Not all curves can be written as y = f(x). When x and y are related implicitly, we differentiate both sides of the equation with respect to x and apply the chain rule to every term involving y. Since y is a function of x, d/dx (y²) = 2y dy/dx.

并非所有曲线都能写成 y = f(x)。当 x 与 y 隐式相关时，我们对等式两边关于 x 求导，并对每个含 y 的项应用链式法则。因为 y 是 x 的函数，所以 d/dx (y²) = 2y dy/dx。

Consider x² + y² = 25. Differentiating gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. This is the gradient of the tangent to a circle of radius 5 at any point (x, y) on the circle.

考虑 x² + y² = 25。求导得 2x + 2y dy/dx = 0，所以 dy/dx = −x/y。这就是半径 5 的圆上任意一点 (x, y) 处切线的斜率。

12. Applications of Differentiation | 微分的应用

Differentiation is applied in many contexts: finding tangents and normals, locating stationary points, solving optimisation problems, and calculating rates of change. Let us summarise the key formulas.

微分在許多背景下都有应用：求切线与法线、寻找驻点、解决最优化问题以及计算变化率。下面总结关键公式。

The equation of the tangent at (x₁, y₁) is y − y₁ = m(x − x₁), where m = dy/dx at that point. The normal is perpendicular, so its gradient is −1/m.

在 (x₁, y₁) 处切线的方程是 y − y₁ = m(x − x₁)，其中 m 是该点的 dy/dx。法线与切线垂直，因此其斜率为 −1/m。

Stationary points: solve dy/dx = 0.
Test for maximum or minimum: use d²y/dx² or the sign change of dy/dx.
Rates of change: if y depends on x and x depends on t, then dy/dt = dy/dx × dx/dt.

驻点：求解 dy/dx = 0。

判断极大或极小值：使用 d²y/dx² 或 dy/dx 的符号变化。

变化率：若 y 依赖于 x 且 x 依赖于 t，则 dy/dt = dy/dx × dx/dt。

For example, the area A of a circle is πr². If the radius is increasing at 2 cm/s, then dA/dt = (dA/dr)(dr/dt) = 2πr × 2 = 4πr cm²/s. When r = 3, the area increases at 12π cm²/s.

例如，圆的面积 A = πr²。若半径以 2 cm/s 的速度增加，则 dA/dt = (dA/dr)(dr/dt) = 2πr × 2 = 4πr cm²/s。当 r = 3 时，面积以 12π cm²/s 的速率增加。

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A-Level Mathematics: Differentiation and Basic Rules of Differentiation | A-Level 数学：微分法与基本求导法则