A-Level Maths: Differentiation – Rules, Techniques & Applications | A-Level数学：微分法则、技巧与应用

Introduction: Why Differentiation Matters

Differentiation is one of the two pillars of calculus, alongside integration. It is the mathematical tool that allows us to measure how one quantity changes in relation to another – in other words, it gives us the rate of change. From calculating the velocity of a moving object to determining the marginal cost in economics, differentiation is fundamental to virtually every quantitative discipline. For A-Level Mathematics students, mastering differentiation is not just about passing exams; it is about developing the analytical mindset that underpins all of applied mathematics, physics, and engineering.

微分是微积分的两大支柱之一，另一个是积分。它是一种数学工具，使我们能够衡量一个量相对于另一个量的变化–换句话说，它给出了变化率。从计算运动物体的速度到确定经济学中的边际成本，微分几乎对每一个定量学科都是基础性的。对于A-Level数学学生来说，掌握微分不仅是为了通过考试；更是为了培养支撑所有应用数学、物理学和工程学的分析思维。

1. The Limit Definition: Differentiation from First Principles

Differentiation begins with a single powerful idea: the derivative of a function at a point is the slope of the tangent line to the curve at that point. More formally, the derivative of f(x) with respect to x is defined as:

f'(x) = lim_h->0 [f(x+h) − f(x)] / h

This expression represents the limiting value of the gradient of a chord as the two points on the curve become infinitesimally close together. When h approaches zero, the secant line through (x, f(x)) and (x+h, f(x+h)) becomes the tangent line at x. This is the formal definition from first principles, and while exam boards rarely ask you to differentiate complicated functions this way, understanding it is essential for grasping what differentiation actually means – not just how to perform it mechanically.

微分始于一个强大的思想：函数在某点的导数就是曲线在该点处切线的斜率。更正式地说，f(x)关于x的导数定义为上述极限表达式。这个表达式表示当曲线上的两点无限接近时弦的梯度的极限值。当h趋近于零时，通过(x, f(x))和(x+h, f(x+h))的割线变成x处的切线。这是从第一原理出发的形式定义，虽然考试很少要求你用这种方式对复杂函数求导，但理解它对于把握微分的真正含义–而不仅仅是机械地执行–至关重要。

Example: Differentiating x² from First Principles

Let f(x) = x². Then:

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

This confirms the familiar rule: the derivative of x² is 2x. Every differentiation rule you learn in A-Level ultimately derives from this fundamental limit process.

2. Basic Differentiation Rules

Once you understand the concept, you can apply standard rules to differentiate efficiently. These rules form the foundation for all more advanced techniques.

一旦理解了概念，你就可以应用标准规则来高效地求导。这些规则构成了所有更高级技术的基础。

The Power Rule

For any function of the form f(x) = xⁿ, where n is a real number:

d/dx [xⁿ] = n·xⁿ⁻¹

This is the single most important differentiation rule. Examples: d/dx [x³] = 3x², d/dx [x¹⁰] = 10x⁹, d/dx [x] = 1·x⁰ = 1, and d/dx [√x] = d/dx [x¹/²] = (1/2)x⁻¹/² = 1/(2√x). The power rule works for any real exponent, including negative and fractional powers.

这是最重要的求导规则。例如：d/dx [x³] = 3x²，d/dx [√x] = 1/(2√x)。幂法则适用于任何实数指数，包括负指数和分数指数。

Constant Multiple Rule

d/dx [c·f(x)] = c·f'(x), where c is a constant. The constant factor simply stays with the derivative. Example: d/dx [5x⁴] = 5·4x³ = 20x³.

Sum and Difference Rule

d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Derivatives distribute across addition and subtraction. Example: d/dx [3x² + 2x − 7] = 6x + 2. Note that the derivative of any constant term is zero – constants have zero rate of change.

导数在加法和减法上可以分配。注意常数项的导数为零–常数没有变化率。

3. The Chain Rule

The chain rule is used to differentiate composite functions – functions of functions. If y = f(g(x)), then:

dy/dx = f'(g(x)) · g'(x)

In Leibniz notation: dy/dx = (dy/du) · (du/dx), where u = g(x). The intuitive idea is that the rate of change of y with respect to x equals the rate of change of y with respect to u multiplied by the rate of change of u with respect to x.

链式法则用于求复合函数的导数。如果y = f(g(x))，则dy/dx = f'(g(x)) · g'(x)。直观的理解是：y关于x的变化率等于y关于u的变化率乘以u关于x的变化率。

Examples of the Chain Rule

Example 1: y = (3x + 2)⁵. Let u = 3x + 2, then y = u⁵. dy/du = 5u⁴, du/dx = 3. So dy/dx = 5(3x+2)⁴ · 3 = 15(3x+2)⁴.

Example 2: y = sin(2x). Let u = 2x, then y = sin(u). dy/du = cos(u), du/dx = 2. So dy/dx = 2cos(2x).

Example 3: y = e^(x²). Let u = x², then y = e^u. dy/du = e^u, du/dx = 2x. So dy/dx = 2x·e^(x²).

With practice, you will learn to apply the chain rule in one step without explicitly writing the substitution: “differentiate the outer function, keep the inner function, multiply by the derivative of the inner function.”

通过练习，你将学会一步应用链式法则而无需显式写出代换：”对外层函数求导，保留内层函数，乘以内层函数的导数。”

4. The Product Rule

When differentiating the product of two functions, y = f(x)·g(x):

dy/dx = f'(x)·g(x) + f(x)·g'(x)

A common mnemonic: “first times derivative of second, plus second times derivative of first” – but be careful with ordering; the formula above is the standard presentation.

当求两个函数乘积的导数时，y = f(x)·g(x)：dy/dx = f'(x)·g(x) + f(x)·g'(x)。常见记忆口诀：”第一项乘以第二项的导数，加上第二项乘以第一项的导数”。

Example

y = x²·sin(x). Let f(x) = x² (f'(x) = 2x) and g(x) = sin(x) (g'(x) = cos(x)). Then dy/dx = 2x·sin(x) + x²·cos(x).

5. The Quotient Rule

When differentiating a quotient, y = f(x)/g(x):

dy/dx = [f'(x)·g(x) − f(x)·g'(x)] / [g(x)]²

The mnemonic for the quotient rule: “low d-high minus high d-low, over low squared” – where “low” is the denominator g(x) and “high” is the numerator f(x). Pay careful attention to the minus sign in the numerator; the order matters.

商的求导法则记忆口诀：”分母乘以分子的导数，减去分子乘以分母的导数，除以分母的平方”。注意分子中的减号–顺序很重要。

Example

y = x / (x+1). Let f(x) = x (f'(x) = 1) and g(x) = x+1 (g'(x) = 1). Then dy/dx = [1·(x+1) − x·1] / (x+1)² = (x+1−x)/(x+1)² = 1/(x+1)².

6. Derivatives of Trigonometric Functions

The standard derivatives of the three main trigonometric functions must be memorised:

• d/dx [sin(x)] = cos(x)
• d/dx [cos(x)] = −sin(x) – note the negative sign!
• d/dx [tan(x)] = sec²(x)

These can be combined with the chain rule for more complex arguments. For example, d/dx [sin(3x)] = 3cos(3x) and d/dx [cos(5x²)] = −10x·sin(5x²).

三个主要三角函数的标准导数必须记住。它们可以与链式法则结合用于更复杂的参数。

Derivatives of Reciprocal Trig Functions

d/dx [sec(x)] = sec(x)tan(x), d/dx [csc(x)] = −csc(x)cot(x), d/dx [cot(x)] = −csc²(x). These are less commonly examined at A-Level but are worth knowing for completeness, particularly for students taking Further Mathematics.

d/dx [sec(x)] = sec(x)tan(x)，d/dx [csc(x)] = −csc(x)cot(x)，d/dx [cot(x)] = −csc²(x)。这些在A-Level考试中不常见，但为了完整性值得了解，特别是对于学习进阶数学的学生。

7. Derivatives of Exponential and Logarithmic Functions

The exponential function e^x is unique: it is its own derivative.

d/dx [e^x] = e^x

For a general exponential: d/dx [a^x] = a^x · ln(a). For the natural logarithm: d/dx [ln(x)] = 1/x, for x > 0. Using the chain rule: d/dx [ln(f(x))] = f'(x)/f(x).

指数函数e^x是独一无二的：它等于它自己的导数。对于自然对数：d/dx [ln(x)] = 1/x (x > 0)。使用链式法则：d/dx [ln(f(x))] = f'(x)/f(x)。

Examples

d/dx [e^(3x)] = 3e^(3x). d/dx [ln(2x+1)] = 2/(2x+1). d/dx [2^x] = 2^x · ln(2).

8. Second Derivatives and Higher Orders

The second derivative, denoted f”(x) or d²y/dx², is the derivative of the derivative. It measures the rate of change of the rate of change – in physics, it represents acceleration. Higher-order derivatives continue this pattern: f”'(x) is the third derivative, f⁽⁴⁾(x) is the fourth, and so on.

二阶导数，记作f”(x)或d²y/dx²，是导数的导数。它衡量变化率的变化率–在物理学中，它表示加速度。更高阶导数继续这一模式。

Example

y = x³ − 6x² + 9x + 1. First derivative: dy/dx = 3x² − 12x + 9. Second derivative: d²y/dx² = 6x − 12.

Second derivatives are crucial for classifying stationary points. If f'(a) = 0 and f”(a) > 0, then x = a is a local minimum. If f'(a) = 0 and f”(a) < 0, then x = a is a local maximum. If f''(a) = 0, the test is inconclusive and you must examine the sign of f'(x) either side of x = a.

二阶导数对于分类驻点至关重要。如果f'(a) = 0且f”(a) > 0，则x = a是局部极小值点。如果f'(a) = 0且f”(a) < 0，则x = a是局部极大值点。如果f''(a) = 0，则检验无法确定，你必须检查x = a两侧f'(x)的符号。

9. Applications of Differentiation

Finding Tangents and Normals

The equation of the tangent line to the curve y = f(x) at the point (a, f(a)) is:

y − f(a) = f'(a)(x − a)

The normal is perpendicular to the tangent. Its gradient is m = −1/f'(a), so the normal equation is:

y − f(a) = −(1/f'(a))(x − a)

曲线y = f(x)在点(a, f(a))处的切线方程为y − f(a) = f'(a)(x − a)。法线与切线垂直，其斜率为m = −1/f'(a)。

Stationary Points and Curve Sketching

Stationary points occur where f'(x) = 0. There are three types: local maxima (gradient changes from positive to negative), local minima (gradient changes from negative to positive), and points of inflection where the gradient has the same sign on both sides. Understanding stationary points allows you to sketch curves accurately, identifying where the function increases, decreases, and changes concavity.

驻点出现在f'(x) = 0处。有三种类型：局部极大值（梯度从正变负）、局部极小值（梯度从负变正）以及拐点（梯度两侧符号相同）。理解驻点使你能准确绘制曲线草图。

Optimisation Problems

Optimisation is one of the most practical applications of differentiation. These problems ask you to find the maximum or minimum value of some quantity subject to given constraints. The general approach: (1) Express the quantity to be optimised as a function of one variable; (2) Differentiate and set the derivative equal to zero; (3) Solve to find stationary points; (4) Use the second derivative test or sign analysis to confirm whether each stationary point is a maximum or minimum; (5) Check endpoints if the domain is restricted.

优化是微分最实际的应用之一。一般方法：(1) 将要优化的量表示为一个变量的函数；(2) 求导并令导数等于零；(3) 解出驻点；(4) 使用二阶导数检验或符号分析确认驻点是极大值还是极小值；(5) 如果定义域有限制，检查端点。

Example Optimisation Problem

A rectangular field of area 100 m² is to be enclosed by a fence. Find the dimensions that minimise the length of fencing required. Let the width be x and the length be y. Given xy = 100, so y = 100/x. The perimeter P = 2x + 2y = 2x + 200/x. Differentiate: dP/dx = 2 − 200/x². Set dP/dx = 0: 2 = 200/x², so x² = 100, x = 10. Check: d²P/dx² = 400/x³ > 0 for x = 10, confirming a minimum. The optimal dimensions are 10 m by 10 m – a square is the most efficient rectangular shape in terms of perimeter for a given area.

一个面积为100平方米的矩形场地需要用围栏围起来。求使围栏长度最小的尺寸。最终得出最优尺寸为10米乘10米–对于给定面积，正方形是周长效率最高的矩形形状。

10. Implicit Differentiation

Not all relationships between x and y can be expressed as y = f(x). Implicit differentiation handles equations where y appears mixed in with x, such as x² + y² = 25. The technique: differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to y terms – each time you differentiate a y term, multiply by dy/dx.

并非所有x和y之间的关系都能表示为y = f(x)。隐函数求导处理y与x混合出现的方程。技术：对两边关于x求导，将y视为x的函数，对y项应用链式法则–每次对y项求导时乘以dy/dx。

Example: Implicit Differentiation of a Circle

Given x² + y² = 25. Differentiating both sides with respect to x: 2x + 2y·(dy/dx) = 0. Solving for dy/dx: 2y·(dy/dx) = −2x, so dy/dx = −x/y. This gives the gradient of the tangent at any point on the circle – notice that at the top of the circle (x = 0, y = 5) the gradient is 0 (horizontal tangent), and at the rightmost point (x = 5, y = 0) the gradient is undefined (vertical tangent).

给定x² + y² = 25。两边关于x求导：2x + 2y·(dy/dx) = 0。解出dy/dx = −x/y。这给出了圆上任意点处切线的梯度。

11. Parametric Differentiation

When a curve is defined parametrically as x = f(t), y = g(t), the derivative dy/dx is found using:

dy/dx = (dy/dt) / (dx/dt) = g'(t) / f'(t)

This is simply the chain rule in another form. For the second derivative in parametric form: d²y/dx² = d/dt (dy/dx) / (dx/dt).

当曲线以参数形式定义时，dy/dx = (dy/dt) / (dx/dt) = g'(t) / f'(t)。这不过是链式法则的另一种形式。

Example

x = t², y = 2t. Then dx/dt = 2t, dy/dt = 2, so dy/dx = 2/(2t) = 1/t. For the second derivative: d/dt (dy/dx) = d/dt (1/t) = −1/t², so d²y/dx² = (−1/t²) / (2t) = −1/(2t³).

12. Exam Tips and Common Mistakes

1. Never forget the chain rule. When differentiating composite functions like sin(3x), e^(5x), or ln(x²+1), always multiply by the derivative of the inner function. This is the single most common error in A-Level differentiation.

当对复合函数求导时，始终要乘以内层函数的导数。这是A-Level微分中最常见的错误。

2. Watch your signs. The derivative of cos(x) is −sin(x), not sin(x). Many marks are lost on sign errors, especially with the quotient rule and when differentiating trigonometric functions.

注意符号。cos(x)的导数是−sin(x)。许多分数因符号错误而丢失，尤其是在使用商法则和求三角函数的导数时。

3. Distinguish between the product rule and the chain rule. If you see multiplication of two separate functions of x (e.g., x²sinx), use the product rule. If you see a function inside another function (e.g., sin(x²)), use the chain rule.

区分乘积法则和链式法则。如果你看到两个独立的x函数相乘（如x²sinx），使用乘积法则。如果你看到一个函数内部嵌套另一个函数（如sin(x²)），使用链式法则。

4. Simplify before differentiating when possible. Expressions like (x²+1)(x²−1) can be expanded to x⁴−1 before differentiating – this is often simpler than applying the product rule.

在可能的情况下先化简再求导。像(x²+1)(x²−1)这样的表达式可以展开为x⁴−1后再求导–这通常比应用乘积法则更简单。

5. Check your answers. When you find dy/dx, try evaluating it at a simple point and see if the result makes physical sense. If a function is increasing, its derivative should be positive. If you are modelling a real-world scenario, check the units.

检查你的答案。当你求出dy/dx时，尝试在一个简单点处计算其值，看看结果是否合理。如果函数在递增，其导数应为正。如果你在建模现实场景，检查单位。

6. For optimisation problems: Always verify that your critical point actually gives the maximum or minimum required – use the second derivative test or the first derivative sign test. Also, ensure you answer the question asked: if the question asks for the maximum volume, give the volume, not just the dimensions.

对于优化问题：始终验证你的临界点确实给出了所需的最大值或最小值。另外，确保你回答了所问的问题。

Differentiation is a skill that improves dramatically with practice. Work through a variety of problems – from pure differentiation drills to applied optimisation and curve sketching – and you will develop both speed and accuracy. Remember: the rules are tools, but understanding is what makes you a mathematician, not just a calculator.

微分是一项通过大量练习可以显著提高的技能。解决各种各样的问题–从纯求导练习到应用优化和曲线草图–你将同时提高速度和准确性。记住：规则是工具，但理解才是使你成为数学家而不是计算器的关键。

Key Terminology / 关键术语表

English	中文
Derivative	导数
Differentiation	微分 / 求导
First Principles	第一原理
Chain Rule	链式法则
Product Rule	乘积法则
Quotient Rule	商法则
Stationary Point	驻点
Turning Point	拐点 / 转折点
Point of Inflection	拐点
Tangent	切线
Normal	法线
Second Derivative	二阶导数
Implicit Differentiation	隐函数求导
Parametric Differentiation	参数方程求导
Optimisation	最优化
Rate of Change	变化率