A-Level Mathematics: Definition and Computation of Derivatives | A-Level 数学：导数的定义与求法

📚 A-Level Mathematics: Definition and Computation of Derivatives | A-Level 数学：导数的定义与求法

The derivative is one of the most powerful concepts in A-Level Mathematics. It measures how a function changes as its input changes, and it lies at the heart of calculus. In this article, we will explore the formal definition of the derivative, the standard rules for computing derivatives, and the key applications that appear in Edexcel A-Level exams.

导数是 A-Level 数学中最核心的概念之一。它衡量函数随自变量变化的速率，也是微积分的基础。本文将系统讲解导数的严格定义、常用求导法则，以及 Edexcel A-Level 考试中的典型应用。

1. The Definition of the Derivative from First Principles | 导数的定义（一阶原理）

The derivative of a function \(f(x)\) at a point \(x\) is defined by the limit of the average rate of change over an interval as that interval shrinks to zero. This is called differentiating from first principles. The formal expression is:

函数 \(f(x)\) 在点 \(x\) 处的导数定义为：当区间长度趋近于零时，函数在该区间上平均变化率的极限。这称为“用一阶原理求导”。其正式表达式为：

f'(x) = lim_h→0 [f(x+h) − f(x)] / h

If this limit exists, the function is said to be differentiable at \(x\). Geometrically, the derivative gives the slope of the tangent line to the curve at that point.

如果此极限存在，则称函数在 \(x\) 处可导。从几何上看，导数表示曲线在该点处切线的斜率。

For example, for \(f(x) = x^2\):

例如，对于 \(f(x) = x^2\)：

f'(x) = lim_h→0 [(x+h)² − x²] / h = lim_h→0 [2xh + h²] / h = lim_h→0 (2x + h) = 2x

Thus the derivative of \(x^2\) is \(2x\). This method is a common first question in Edexcel papers, especially for quadratic or cubic functions.

因此 \(x^2\) 的导数是 \(2x\)。这种方法在 Edexcel 试卷中经常出现，尤其是针对二次或三次函数。

2. Notation and Basic Interpretation | 导数记号与基本含义

There are several standard notations for the derivative. If \(y = f(x)\), then the derivative can be written as \(f'(x)\), \(y’\), or \(\frac{dy}{dx}\). All of these mean the same thing. The notation \(\frac{dy}{dx}\) reminds us that the derivative is a rate of change of \(y\) with respect to \(x\).

导数的记号有多种。若 \(y = f(x)\)，导数可记为 \(f'(x)\)、\(y’\) 或 \(\frac{dy}{dx}\)，它们含义相同。\(\frac{dy}{dx}\) 提醒我们导数表示 \(y\) 关于 \(x\) 的变化率。

For example, if \(s(t)\) is the position of an object at time \(t\), then \(s'(t)\) is its velocity. In exam contexts you must be comfortable converting between these notations, especially when dealing with kinematics problems.

例如，若 \(s(t)\) 表示物体在时刻 \(t\) 的位置，则 \(s'(t)\) 就是速度。在考试中，你需要熟练转换各种记号，尤其在运动学问题中。

Key interpretations to remember:

需要牢记的关键含义：

The derivative is a function that gives the slope of the tangent at any point. 导数是给出任意点切线斜率的函数。
The sign of the derivative indicates whether the function is increasing or decreasing. 导数的正负表示函数递增还是递减。
The derivative can be evaluated at a specific point, such as \(f'(2)\). 导数可以在特定点求值，例如 \(f'(2)\)。

3. Differentiating Polynomials Using the Power Rule | 幂法则求多项式导数

The power rule is the most frequently used differentiation rule. For any constant \(n\), the derivative of \(x^n\) is \(n x^{n-1}\). This works for positive, negative, and fractional powers.

幂法则是最常用的求导法则。对任意常数 \(n\)，\(x^n\) 的导数为 \(n x^{n-1}\)。该法则适用于正指数、负指数和分数指数。

If y = xⁿ, then dy/dx = n xⁿ⁻¹

In addition, the derivative of a constant is zero, and constant multiples are preserved. For example:

此外，常数的导数为零，常数倍保持不变。例如：

y = 3x⁴ − 5x² + 7 ⇒ dy/dx = 12x³ − 10x

Notice that the constant term 7 disappears. This is because the derivative of a constant is zero, and the derivative represents a rate of change, which a constant value does not affect.

注意常数项 7 消失了。这是因为常数的导数为零，而导数代表变化率，常数项不会影响变化率。

4. Product and Quotient Rules | 乘积法则与商法则

When a function is written as a product of two functions, we use the product rule. If \(y = u v\), then:

当一个函数是两个函数相乘时，我们使用乘积法则。若 \(y = u v\)，则：

dy/dx = u (dv/dx) + v (du/dx)

For example, if \(y = x² \sin x\), then letting \(u = x²\) and \(v = \sin x\), we get:

例如，若 \(y = x² \sin x\)，令 \(u = x²\)，\(v = \sin x\)，则：

dy/dx = x² cos x + 2x sin x

When a function is written as a quotient of two functions, we use the quotient rule. If \(y = u/v\), then:

当一个函数是两个函数相除时，我们使用商法则。若 \(y = u/v\)，则：

dy/dx = [v (du/dx) − u (dv/dx)] / v²

For example, for \(y = x / (x + 1)\):

例如，对于 \(y = x / (x + 1)\)：

dy/dx = [1·(x + 1) − x·1] / (x + 1)² = 1 / (x + 1)²

5. The Chain Rule | 链式法则

The chain rule is used for composite functions. If \(y\) is a function of \(u\), and \(u\) is a function of \(x\), then:

链式法则用于复合函数。若 \(y\) 是 \(u\) 的函数，而 \(u\) 是 \(x\) 的函数，则：

dy/dx = (dy/du) × (du/dx)

For example, to differentiate \(y = (3x + 2)⁵\), let \(u = 3x + 2\). Then \(y = u⁵\), so \(dy/du = 5u⁴\) and \(du/dx = 3\). Therefore:

例如，求 \(y = (3x + 2)⁵\) 的导数，令 \(u = 3x + 2\)，则 \(y = u⁵\)，所以 \(dy/du = 5u⁴\)，\(du/dx = 3\)。因此：

dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴

The chain rule can be extended to nested functions. In Edexcel exam questions, you often need to combine the chain rule with the product and quotient rules. The key is to identify the “inner” and “outer” functions clearly before differentiating.

链式法则可推广到复合多层函数。在 Edexcel 考试题中，经常需要将链式法则与乘积法则、商法则结合使用。关键在于求导前清晰识别“内层”和“外层”函数。

6. Derivatives of Exponential and Logarithmic Functions | 指数函数和对数函数的导数

The derivative of \(e^x\) is itself: \(e^x\). More generally, for a constant \(a\), the derivative of \(e^{kx}\) is \(k e^{kx}\), using the chain rule.

\(e^x\) 的导数是它本身：\(e^x\)。更一般地，对于常数 \(a\)，\(e^{kx}\) 的导数为 \(k e^{kx}\)，这需要用到链式法则。

d/dx (eˣ) = eˣ, d/dx (e^{kx}) = k e^{kx}

For logarithms, the derivative of \(\ln x\) is \(1/x\). For \(\ln(f(x))\), we use the chain rule to get \(f'(x)/f(x)\).

对于对数函数，\(\ln x\) 的导数是 \(1/x\)。对于 \(\ln(f(x))\)，使用链式法则得到 \(f'(x)/f(x)\)。

For example:

例如：

y = ln(2x + 1) ⇒ dy/dx = 2 / (2x + 1)

These rules appear frequently in Edexcel pure mathematics papers. In particular, examiners often ask you to differentiate exponential expressions combined with polynomials.

这些法则在 Edexcel 纯数试卷中频繁出现。特别是，出题人经常要求你求指数表达式与多项式组合的导数。

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

The derivatives of the basic trigonometric functions must be memorised. If angles are measured in radians, then:

基本三角函数的导数必须牢记。当角度以弧度制表示时：

d/dx (sin x) = cos x | d/dx (sin x) = cos x
d/dx (cos x) = −sin x | d/dx (cos x) = −sin x
d/dx (tan x) = sec² x | d/dx (tan x) = sec² x

For composite trigonometric functions, use the chain rule. For example:

对于复合三角函数，使用链式法则。例如：

y = sin(3x) ⇒ dy/dx = 3 cos(3x)

For products involving trigonometric functions, you may need the product rule. For example, \(y = x \cos x\):

对于包含三角函数的乘积，可能用到乘积法则。例如 \(y = x \cos x\)：

dy/dx = cos x − x sin x

Remember to leave answers in terms of trigonometric functions where appropriate, and do not convert to degrees unless the question specifies.

请记住，答案中保留三角函数形式，除非题目特别说明，不要转换为角度制。

8. Tangents and Normals | 切线与法线

One major application of the derivative is finding equations of tangents and normals to a curve at a given point. If \(y = f(x)\), then the gradient of the tangent at \(x = a\) is \(m = f'(a)\). The equation of the tangent is:

导数的一大应用是求曲线在某一点处的切线和法线方程。若 \(y = f(x)\)，则曲线在 \(x = a\) 处切线的斜率为 \(m = f'(a)\)。切线方程为：

y − f(a) = f'(a)(x − a)

The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent gradient, provided \(f'(a) \neq 0\). Thus the normal equation is:

法线与切线垂直，因此它的斜率是切线斜率的负倒数（前提是 \(f'(a) \neq 0\)）。因此法线方程为：

y − f(a) = −1 / f'(a) (x − a)

For example, find the tangent to \(y = x²\) at \(x = 3\). Since \(f'(x) = 2x\), we have \(f'(3) = 6\) and \(f(3) = 9\). The tangent is \(y − 9 = 6(x − 3)\), which simplifies to \(y = 6x − 9\).

例如，求曲线 \(y = x²\) 在 \(x = 3\) 处的切线。由于 \(f'(x) = 2x\)，得 \(f'(3) = 6\)，\(f(3) = 9\)。切线为 \(y − 9 = 6(x − 3)\)，化简得 \(y = 6x − 9\)。

9. Stationary Points and Increasing/Decreasing Functions | 驻点与函数的增减性

A stationary point occurs where the derivative is zero, i.e. \(f'(x) = 0\). At such points, the tangent is horizontal. Stationary points can be local maxima, local minima, or points of inflexion.

驻点出现于导数为零之处，即 \(f'(x) = 0\)。在驻点处，切线水平。驻点可以是局部最大值、局部最小值或拐点。

To determine the nature of a stationary point, examine the sign of the derivative just to the left and right of the point. Alternatively, use the second derivative test:

为判断驻点的性质，可以检查驻点左右两侧导数的符号。也可以使用二阶导数检验：

If \(f”(x) > 0\), the point is a local minimum. 若 \(f”(x) > 0\)，则该点为局部最小值。
If \(f”(x) < 0\), the point is a local maximum. 若 \(f”(x) < 0\)，则该点为局部最大值。
If \(f”(x) = 0\), the test is inconclusive; use the sign change method. 若 \(f”(x) = 0\)，检验失效，需使用符号变化法。

Functions are increasing where \(f'(x) > 0\) and decreasing where \(f'(x) < 0\). These intervals are often asked as part of a longer question in Edexcel papers.

函数在 \(f'(x) > 0\) 时递增，在 \(f'(x) < 0\) 时递减。Edexcel 试卷经常在综合题中要求求出这些区间。

10. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative, denoted \(f”(x)\) or \(d²y/dx²\), is the derivative of the derivative. It measures the rate of change of the gradient and provides information about the curvature of the graph.

二阶导数记为 \(f”(x)\) 或 \(d²y/dx²\)，是导数的导数。它衡量斜率的变化率，并揭示函数图像的弯曲情况。

If \(f”(x) > 0\) on an interval, the graph is concave up (smiling shape). If \(f”(x) < 0\), the graph is concave down (frowning shape).

若在某个区间上 \(f”(x) > 0\)，则图像上凹（开口向上）。若 \(f”(x) < 0\)，则图像下凹（开口向下）。

A point where the concavity changes is called a point of inflexion. At a point of inflexion, \(f”(x) = 0\), but this condition alone is not sufficient; the sign of \(f”(x)\) must change on either side.

凹凸性发生改变的点称为拐点。在拐点处 \(f”(x) = 0\)，但仅有此条件不够，还必须满足 \(f”(x)\) 在两侧变号。

For example, for \(y = x³\), \(y’ = 3x²\) and \(y” = 6x\). At \(x = 0\), \(y” = 0\), and the concavity changes from down to up. Therefore \((0,0)\) is a point of inflexion.

例如，对于 \(y = x³\)，\(y’ = 3x²\)，\(y” = 6x\)。在 \(x = 0\) 处，\(y” = 0\)，凹凸性由下凹变为上凹。因此 \((0,0)\) 是拐点。

11. Practical Examples and Exam Tips | 实际例题与考试提示

Let us work through a typical Edexcel-style question. Given \(f(x) = x³ − 6x² + 9x\), find the stationary points and classify them.

让我们完成一道典型的 Edexcel 风格题目。已知 \(f(x) = x³ − 6x² + 9x\)，求驻点并判断其性质。

First, differentiate:

首先求导：

f'(x) = 3x² − 12x + 9

Set \(f'(x) = 0\):

令 \(f'(x) = 0\)：

3(x² − 4x + 3) = 0 ⇒ 3(x − 1)(x − 3) = 0

Thus \(x = 1\) and \(x = 3\). Now evaluate \(f”(x) = 6x − 12\). At \(x = 1\), \(f”(1) = −6 < 0\), so \((1,4)\) is a local maximum. At \(x = 3\), \(f”(3) = 6 > 0\), so \((3,0)\) is a local minimum.

因此 \(x = 1\) 和 \(x = 3\)。现在计算 \(f”(x) = 6x − 12\)。在 \(x = 1\) 处，\(f”(1) = −6 < 0\)，所以 \((1,4)\) 是局部最大值。在 \(x = 3\) 处，\(f”(3) = 6 > 0\)，所以 \((3,0)\) 是局部最小值。

Exam tips for differentiation:

求导部分的考试建议：

Always rewrite expressions as powers of \(x\) before differentiating. 求导前先将表达式写成 \(x\) 的幂形式。
Simplify fractions and negative exponents carefully. 仔细化简分数和负指数。
Check whether the question asks for exact values or decimal approximations. 确认题目要求精确值还是近似值。
Use radians for trigonometric derivatives unless stated otherwise. 如无特殊说明，三角函数的导数使用弧度制。
For worded problems, clearly state the units of the derivative where applicable. 在应用题中，如适用，注明导数的单位。

By mastering the definition of the derivative and practising the standard rules, you will be well prepared for any differentiation question in your A-Level exam.

通过掌握导数的定义并练习标准求导法则，你将能够从容应对 A-Level 考试中的任何求导问题。

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A-Level Mathematics: Definition and Computation of Derivatives | A-Level 数学：导数的定义与求法