AS Mathematics: Differentiation Essentials | AS 数学：微分考点精讲

📚 AS Mathematics: Differentiation Essentials | AS 数学：微分考点精讲

Differentiation is one of the two cornerstones of calculus and forms a major part of AS Mathematics. It allows us to find the rate at which one quantity changes with respect to another, most commonly the gradient of a curve at any point. In this revision guide we cover every essential topic: from first principles through the chain, product and quotient rules, to applications such as tangents, normals, stationary points, and optimisation. Worked examples and clear explanations are paired with Chinese translations to help bilingual learners master the content confidently for the AS exam.

微分是微积分的两大基石之一，也是 AS 数学的重要板块。它帮助我们求解一个量相对于另一个量的变化率，最常见的应用便是曲线在任意点的斜率。本复习指南涵盖所有核心考点：从第一原理到链式法则、积法则和商法则，再到切线、法线、驻点以及优化等应用。每个知识点都配有实例讲解和中英对照，帮助双语学习者牢固掌握 AS 考试内容。

1. What is Differentiation? | 什么是微分？

Differentiation is the process of finding the derivative of a function. The derivative, denoted f'(x) or dy/dx, measures the instantaneous rate of change of y with respect to x. Geometrically, it gives the gradient of the tangent line to the curve y = f(x) at any point x. For a straight line the gradient is constant, but for a curve it varies from point to point, and that is exactly the value the derivative provides.

微分是求函数导数的过程。导数记为 f'(x) 或 dy/dx，它衡量了 y 关于 x 的瞬时变化率。从几何上讲，它给出了曲线 y = f(x) 在任意点 x 处切线的斜率。对于直线，斜率是恒定的，而对于曲线，斜率逐点变化，而导数提供的正是该变化值。

2. Differentiation from First Principles | 从第一原理求导

The formal definition of the derivative uses a limit. If we have a function f(x), its derivative f'(x) is defined as:

f ‘(x) = limₕ→₀ (f(x + h) − f(x)) / h

导数的正式定义借助极限。若函数为 f(x)，其导数 f'(x) 定义为：

f ‘(x) = limₕ→₀ (f(x + h) − f(x)) / h

To use this, we substitute the expression for f(x+h) into the formula, simplify, and then let h approach 0. For example, for f(x)=x², we have f(x+h)=(x+h)²=x²+2xh+h². The difference quotient becomes (2xh+h²)/h = 2x+h, which tends to 2x as h→0. Thus the derivative of x² is 2x. This method underpins all differentiation rules.

使用时，将 f(x+h) 的表达式代入公式，化简后令 h 趋于 0。例如对于 f(x)=x²，f(x+h)=(x+h)²=x²+2xh+h²，差商变为 (2xh+h²)/h = 2x+h，当 h→0 时极限为 2x。因此 x² 的导数为 2x。这一方法是所有求导法则的基础。

3. Basic Differentiation Rules | 基本求导法则

Instead of using first principles every time, we remember the standard derivatives of common functions. For any real constant n, the power rule states:

d/dx (xⁿ) = n xⁿ⁻¹

我们不必每次都从第一原理出发，而是熟记常见函数的标准导数。对任何实数常数 n，幂函数求导法则为：

d/dx (xⁿ) = n xⁿ⁻¹

The derivative of a constant is zero. The derivative of a constant multiplied by a function is the constant times the derivative of the function: d/dx [c·f(x)] = c·f'(x). Sums and differences are differentiated term by term. A typical AS differentiation table looks like:

常数的导数为零。常数乘以函数的导数等于常数乘以函数的导数：d/dx [c·f(x)] = c·f'(x)。和与差可以逐项求导。一个典型的 AS 求导公式表如下：

Function f(x)	Derivative f'(x)
c (constant)	0
x	1
x²	2x
xⁿ	n xⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	− sin x

These rules allow you to differentiate any polynomial and many other functions quickly. For instance, the derivative of 3x⁴ − 5x² + 7 is 12x³ − 10x.

掌握这些法则后，你就能快速地对任意多项式及许多其他函数求导。例如 3x⁴ − 5x² + 7 的导数是 12x³ − 10x。

4. The Product Rule | 积法则

When a function is the product of two simpler functions, y = u(x)·v(x), we cannot simply multiply their individual derivatives. The product rule states:

dy/dx = u·(dv/dx) + v·(du/dx)

当函数是两个较简单函数的乘积时，即 y = u(x)·v(x)，我们不能简单地将各自的导数相乘。积法则指出：

dy/dx = u·(dv/dx) + v·(du/dx)

An easy way to remember it is: ‘first times derivative of second plus second times derivative of first’. For example, if y = x² sin x, let u = x², v = sin x, then u’ = 2x, v’ = cos x. The derivative is x² cos x + sin x · 2x = x² cos x + 2x sin x. Always leave it in a simplified factored form if possible.

一个简单的记法是：“第一乘以第二的导数，加上第二乘以第一的导数”。例如 y = x² sin x，设 u = x²，v = sin x，则 u’ = 2x，v’ = cos x。导数为 x² cos x + sin x · 2x = x² cos x + 2x sin x。尽量保留化简后的因式分解形式。

5. The Quotient Rule | 商法则

If a function is a quotient, y = u(x) / v(x), the derivative is given by the quotient rule:

dy/dx = (v·(du/dx) − u·(dv/dx)) / v²

若函数为商的形式 y = u(x) / v(x)，其导数则由商法则给出：

dy/dx = (v·(du/dx) − u·(dv/dx)) / v²

The mnemonic is: ‘bottom times derivative of top minus top times derivative of bottom, all over bottom squared’. For y = x / (x+1), let u = x, v = x+1, then u’ = 1, v’ = 1. Substituting gives dy/dx = [(x+1)·1 − x·1] / (x+1)² = 1 / (x+1)². Remember to square the denominator carefully and never cancel terms before expanding when necessary.

记忆口诀为：“分母乘以分子的导数，减去分子乘以分母的导数，整体除以分母的平方”。对于 y = x / (x+1)，设 u = x，v = x+1，则 u’ = 1，v’ = 1。代入得 dy/dx = [(x+1)·1 − x·1] / (x+1)² = 1 / (x+1)²。注意分母平方要准确，必要时要先展开再约分。

6. The Chain Rule | 链式法则

The chain rule is used when we have a function of a function, i.e., y = f(g(x)). If we let u = g(x), then y = f(u), and the derivative is:

dy/dx = dy/du × du/dx

处理复合函数（即函数的函数）时需要使用链式法则。设 u = g(x)，则 y = f(u)，其导数为：

dy/dx = dy/du × du/dx

This rule extends naturally to more complicated nests. For AS Level, typical examples include differentiating (3x+1)⁵ or sin(2x). For y = (3x+1)⁵, let u = 3x+1, then y = u⁵. So dy/du = 5u⁴ and du/dx = 3, giving dy/dx = 15(3x+1)⁴. For y = sin(2x), we get cos(2x)·2 = 2 cos(2x). Always look for an ‘inner function’ and an ‘outer function’ and apply the chain rule systematically.

该法则可以自然地推广到更复杂的嵌套函数。在 AS 阶段，典型的例子包括 (3x+1)⁵ 或 sin(2x) 的求导。对于 y = (3x+1)⁵，令 u = 3x+1，则 y = u⁵，dy/du = 5u⁴，du/dx = 3，因此 dy/dx = 15(3x+1)⁴。对于 y = sin(2x)，得到 cos(2x)·2 = 2 cos(2x)。始终要识别出“内层函数”和“外层函数”，并系统地应用链式法则。

7. Second Derivative | 二阶导数

The second derivative is simply the derivative of the derivative. It is denoted f”(x) or d²y/dx². It tells us about the rate of change of the gradient itself, and is crucial for determining the nature of stationary points. To find it, you differentiate the first derivative again using the same rules.

二阶导数就是导数的导数，记为 f”(x) 或 d²y/dx²。它反映了斜率本身的变化率，对于判断驻点的性质至关重要。求取时只需对一阶导数再次使用同样的求导法则即可。

For example, if y = x³ − 3x² + 5x − 7, then dy/dx = 3x² − 6x + 5, and d²y/dx² = 6x − 6. At a particular point, the sign of the second derivative tells us whether the function is concave up (positive) or concave down (negative).

例如 y = x³ − 3x² + 5x − 7，一阶导数 dy/dx = 3x² − 6x + 5，二阶导数 d²y/dx² = 6x − 6。在某一点处，二阶导数的符号表明了函数在该点处是凹向上（正值）还是凹向下（负值）。

8. Equations of Tangents and Normals | 切线与法线方程

Once you have the derivative, you can find the equation of the tangent to a curve at a given point. The tangent line touches the curve at that point and has the same gradient as the derivative there. The normal is perpendicular to the tangent. If the gradient of the tangent is m, the gradient of the normal is −1/m (provided m ≠ 0).

一旦求出导数，就可以求曲线在指定点的切线方程。切线在该点与曲线相切，且其斜率即为该点的导数值。法线则与切线垂直。若切线斜率为 m，则法线斜率为 −1/m（前提是 m ≠ 0）。

If we want the tangent to y = f(x) at x = a, we first find the y-coordinate y₁ = f(a), then the gradient m = f'(a). The equation is then y − y₁ = m(x − a). For a normal, use gradient −1/m instead. For instance, for the curve y = x² at x = 1, the point is (1,1), gradient m = 2, so the tangent is y − 1 = 2(x − 1) → y = 2x − 1. The normal has gradient −1/2, giving y − 1 = −1/2(x − 1) → y = −1/2 x + 3/2.

如果要求曲线 y = f(x) 在 x = a 处的切线，首先求出 y 坐标 y₁ = f(a)，然后求斜率 m = f'(a)。切线方程为 y − y₁ = m(x − a)。法线则只需将斜率替换为 −1/m。例如曲线 y = x² 在 x = 1 处，点为 (1,1)，斜率 m = 2，因此切线为 y − 1 = 2(x − 1) → y = 2x − 1。法线斜率为 −1/2，方程为 y − 1 = −1/2(x − 1) → y = −1/2 x + 3/2。

9. Increasing and Decreasing Functions | 函数的递增与递减

A function is increasing on an interval if its derivative is positive there, and decreasing if its derivative is negative. This sign analysis of dy/dx helps us understand the shape of the graph without plotting it. To find intervals of increase or decrease, solve f'(x) > 0 or f'(x) < 0.

如果函数在某个区间内的导数为正，则函数在该区间递增；若导数为负，则函数递减。通过对 dy/dx 的符号分析，我们无需画图就能了解图像的形状。要求递增或递减区间，只需解不等式 f'(x) > 0 或 f'(x) < 0。

For example, for f(x) = x³ − 3x, we have f'(x) = 3x² − 3 = 3(x² − 1). Setting f'(x) = 0 gives x = −1, 1. Testing sign regions: for x < −1, f'(x) positive (increasing); for −1 < x < 1, f'(x) negative (decreasing); for x > 1, f'(x) positive (increasing). Always show the critical values and a sign diagram in your working.

例如对于 f(x) = x³ − 3x，有 f'(x) = 3x² − 3 = 3(x² − 1)。令 f'(x) = 0 解得 x = −1, 1。检测符号区间：当 x < −1 时 f'(x) 为正（递增）；当 −1 < x < 1 时 f'(x) 为负（递减）；当 x > 1 时 f'(x) 为正（递增）。解题时务必标出临界值并画出符号图。

10. Stationary Points and Their Nature | 驻点及其性质

Stationary points occur where dy/dx = 0, i.e., where the tangent is horizontal. There are three types: local maxima, local minima, and points of inflection. To classify a stationary point, you can use one of two methods: the second derivative test or the first derivative sign test.

驻点出现在 dy/dx = 0 的地方，即切线为水平线的点。驻点分为三类：局部极大值、局部极小值和拐点。要判定驻点类型，可采用两种方法之一：二阶导数判别法或一阶导数符号判别法。

For y = f(x), find x where f'(x) = 0. Then compute f”(x) at that x. If f”(x) > 0, it is a local minimum (concave up). If f”(x) < 0, it is a local maximum (concave down). If f''(x) = 0, the test is inconclusive and you should check the sign of f'(x) just to the left and right of the point. For instance, for y = x³ − 3x, stationary points at x = 1 (f''(1) = 6 > 0 → minimum) and x = −1 (f”(−1) = −6 < 0 → maximum).

对于 y = f(x)，先解出 f'(x) = 0 对应的 x 值，再计算该点的二阶导数 f”(x)。若 f”(x) > 0，该点为局部极小值（凹向上）；若 f”(x) < 0，则为局部极大值（凹向下）。若 f''(x) = 0，则该方法无效，需检查驻点左、右两侧一阶导数的符号。例如 y = x³ − 3x，驻点 x = 1 处 f''(1) = 6 > 0 → 极小值；x = −1 处 f”(−1) = −6 < 0 → 极大值。

11. Optimisation Problems | 优化问题

One of the most practical applications of differentiation is optimisation – finding the maximum or minimum values of a quantity subject to given constraints. A typical AS problem involves forming an expression for the quantity to be optimised in terms of one variable, differentiating, setting the derivative to zero, and confirming the nature of the stationary point.

微分最实际的应用之一便是优化——在给定约束条件下，求某个量的最大值或最小值。AS 阶段的典型问题包括：将待优化量表示为一个变量的表达式，求导，令导数为零，并确认驻点的性质。

For example, a farmer has 100 m of fencing and wants to enclose a rectangular plot against a straight wall. Let the side perpendicular to the wall be x. Then the side parallel to the wall uses (100 − 2x) m. The area A = x(100 − 2x) = 100x − 2x². Differentiating: dA/dx = 100 − 4x. Set to zero: x = 25. Second derivative d²A/dx² = −4 < 0, so it is a maximum. The maximum area is 25 × 50 = 1250 m². Always check that your answer makes sense in the context.

例如，一个农民要用 100 米长的篱笆，沿一面笔直的墙围出一块矩形地块。设垂直于墙的边长为 x，则平行于墙的边所用篱笆长度为 (100 − 2x) 米。面积 A = x(100 − 2x) = 100x − 2x²。求导得 dA/dx = 100 − 4x，令其为零解得 x = 25。二阶导数 d²A/dx² = −4 < 0，故为极大值。最大面积为 25 × 50 = 1250 平方米。始终要检验答案在具体情境中是否合理。

12. Summary and Exam Tips | 总结与应试技巧

To succeed in AS differentiation questions, you must be fluent in all basic derivative rules, recognise when to apply the product, quotient and chain rules, and be able to interpret derivatives in geometric and physical contexts. Practise writing clear, logical solutions with proper notation. Always show the derivative you are working with, and when solving for stationary points or equation of a tangent, present your steps methodically.

要在 AS 微分题目中取得好成绩，你必须熟练所有基本求导法则，能识别何时使用积法则、商法则和链式法则，并能在几何和实际情境中解释导数的意义。多练习写出清晰、逻辑严密的解答，并使用正确的符号。务必展示你所处理的导数表达式，在求解驻点或切线方程时，步骤要条理分明。

Memorise the derivatives of sin x, cos x, eˣ and ln x. Never forget the constant of integration – wait, that’s for integration! For differentiation, double-check your power rule for fractional and negative indices: √x becomes ½ x⁻½, and 1/x² differentiates to −2 x⁻³. If you get stuck, try simplifying the function before differentiating. Good luck!

牢记 sin x、cos x、eˣ 和 ln x 的导数。千万别把积分常数写进来——那是积分才需要的！做微分时要复查分数指数和负指数的幂法则：√x 的导数为 ½ x⁻½，1/x² 的导数为 −2 x⁻³。如果一时卡住，不妨先将函数化简再求导。祝你好运！

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AS Mathematics: Differentiation Essentials | AS 数学：微分 考点精讲