📚 Differentiation in IB Mathematics | IB数学中的微分
Differentiation is a cornerstone of calculus, enabling us to understand how quantities change. In the IB Mathematics curriculum, you will learn to compute derivatives of polynomial, trigonometric, exponential, and logarithmic functions, apply differentiation rules, and use these skills to solve real-world problems such as optimizing areas or finding maximum profits.
微分是微积分的基石，使我们能够理解量如何变化。在IB数学课程中，你将学习计算多项式、三角函数、指数函数和对数函数的导数，应用微分法则，并利用这些技能解决现实世界中的问题，比如优化面积或求最大利润。
This article follows the syllabus of both Applications and Interpretation (AI) and Analysis and Approaches (AA), at Standard Level and Higher Level. The core rules are shared by all four routes; the HL courses simply extend them to implicit differentiation, related rates and more demanding optimization problems.
本文同时对应“应用与解释”（AI）与“分析与方法”（AA）两个方向，涵盖标准水平与高水平。四条路径共享同一套核心法则；HL课程只是把它们延伸到隐函数微分、相关变化率以及更复杂的优化问题。
1. Introduction to Differentiation | 微分导论
Differentiation is the process of finding the derivative of a function. Geometrically, the derivative at a point represents the slope of the tangent line to the graph of the function at that point. If the function describes displacement, the derivative gives velocity. In economics, it can represent marginal cost or marginal revenue. Understanding the concept of instantaneous rate of change is essential before any rule is memorised.
微分是求函数导数的过程。从几何上看，某一点的导数表示函数图像在该点切线的斜率。如果函数描述位移，导数则给出速度。在经济学中，它可以表示边际成本或边际收益。在背诵任何法则之前，理解瞬时变化率的概念至关重要。
The derivative is a local property: it tells you what happens in an infinitesimally small neighbourhood of a single point, not over a whole interval. That is exactly why it answers questions such as "how fast is the population growing right now?" rather than "how much did it grow this year?"
导数是一种局部性质：它描述的是某一点附近无穷小区间内发生的事情，而不是整个区间上的平均情况。正因如此，它回答的是“人口此刻增长多快”，而不是“今年一共增长了多少”。
The notation used in IB includes f ′(x), y ′, and dy/dx. All are equivalent. The dy/dx notation, introduced by Leibniz, emphasises the rate of change of y with respect to x, and it becomes especially useful when the chain rule or related rates are involved. Newton's dot notation is not required by the IB syllabus.
IB中使用的记号包括 f ′(x)、y ′ 和 dy/dx，它们彼此等价。莱布尼茨引入的 dy/dx 记号强调了 y 相对于 x 的变化率，在涉及链式法则或相关变化率时尤其好用。IB大纲不要求使用牛顿的点记号。
2. Limits and the Definition of the Derivative | 极限与导数定义
Formally, the derivative of a function f at x is defined as the limit of the difference quotient as h approaches zero. This is the "first principles" definition and it appears regularly in Paper 1 and in proof-style questions.
形式上，函数 f 在 x 处的导数定义为当 h 趋于零时差商的极限。这就是“第一原理”定义，它在Paper 1和证明类题目中经常出现。
f ′(x) = limh→0 [f(x + h) − f(x)] / h
An equivalent form uses a point a instead of the variable x:
等价的写法是用定点 a 代替变量 x：
f ′(a) = limx→a [f(x) − f(a)] / (x − a)
To differentiate from first principles you follow three steps: write down f(x + h), subtract f(x), and divide by h, then simplify the fraction so that h cancels before taking the limit. Skipping the algebra almost always loses method marks.
用第一原理求导遵循三个步骤：写出 f(x + h)，减去 f(x)，再除以 h，然后化简分式，使 h 在取极限之前被约去。跳过代数化简几乎总是会丢掉方法分。
technique proves that the derivative of x³ is 3x² and that the derivative of 1/x is −1/x².
例如：对于 f(x) = x²，可得 [x² + 2xh + h² − x²] / h = 2x + h，当 h → 0 时趋于 2x。同样的方法可以证明 x³ 的导数是 3x²，1/x 的导数是 −1/x²。
A function is differentiable at a point only if it is continuous there, and the left-hand and right-hand limits of the difference quotient must agree. Corners, cusps and vertical tangents are therefore points where the derivative fails to exist.
只有当函数在某点连续，并且差商的左极限与右极限相等时，函数在该点才可导。因此，尖点、拐锐点与竖直切线都是导数不存在的位置。
3. The Basic Rules: Power, Constant and Sum | 基本法则：幂、常数与和
Once the definition is understood, the working rules make differentiation fast. The power rule states that you multiply by the exponent and then reduce the exponent by one. The constant rule and the sum rule follow from the linearity of the limit process.
理解了定义之后，运算法则能让求导变得快捷。幂法则要求先乘以指数，再把指数减一。常数法则与和法则来自极限运算的线性性质。
n xⁿ⁻¹ d/dx (c) = 0 d/dx [a f(x) ± b g(x)] = a f ′(x) ± b g ′(x)
The rule works for any real exponent, including negative and fractional powers, which is why it is often better to rewrite roots and reciprocals before differentiating:
该法则对任意实数指数都成立，包括负指数与分数指数，因此求导前先改写根式和倒数往往更省事：
- √x = x½, so d/dx (√x) = ½ x−½ = 1 / (2√x) — √x = x½，因此 d/dx (√x) = ½ x−½ = 1 / (2√x)
- 1/x² = x⁻², so d/dx (1/x²) = −2x⁻³ = −2/x³ — 1/x² = x⁻²，因此 d/dx (1/x²) = −2x⁻³ = −2/x³
- 1/x = x⁻¹, so d/dx (1/x) = −x⁻² = −1/x² — 1/x = x⁻¹，因此 d/dx (1/x) = −x⁻² = −1/x²
+ 7. Differentiate term by term, and remember that a constant term always disappears. For AI students this is frequently combined with a GDC check, but the algebraic answer must still be shown in Paper 1.
一个典型的多项式例子：f(x) = 3x⁴ − 5x² + 7x − 4，则 f ′(x) = 12x³ − 10x + 7。逐项求导，并牢记常数项一定会消失。对AI学生来说，这常常需要结合图形计算器验证，但在Paper 1中仍必须写出代数过程。
4. Product, Quotient and Chain Rules | 乘积、商与链式法则
Composite and combined expressions require more than the basic rules. The three advanced rules below cover almost every function you will meet in the IB examinations.
复合与组合形式的表达式需要比基本法则更强的工具。下面三条进阶法则几乎覆盖了IB考试中会遇到的所有函数。
)′ = u′v + uv′ (u / v)′ = (u′v − uv′) / v² dy/dx = dy/du × du/dx
The product rule is symmetric, so the order of the two terms does not matter. The quotient rule, by contrast, is not symmetric: the numerator must be "denominator times derivative of numerator minus numerator times derivative of denominator", and reversing it produces a sign error that examiners look for.
乘积法则是 симметр 的，两项顺序可以互换；商法则则不然：分子必须是“分母乘分子的导数，减去分子乘分母的导数”，一旦颠倒就会出现符号错误，而这正是考官重点排查的地方。
The chain rule is the single most important rule at HL and the most common source of lost marks at SL. In words: differentiate the outside function, keep the inside function unchanged, then multiply by the derivative of the inside.
链式法则是HL中最重要的法则，也是SL最常见的失分点。用语言表述就是：先对外层函数求导，内层函数保持不变，再乘以内层函数的导数。
- y = (2x + 1)⁵ → dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴ — y = (2x + 1)⁵ → dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴
- y = x²eˣ → dy/dx = 2xeˣ + x²eˣ = xeˣ(x + 2) — y = x²eˣ → dy/dx = 2xeˣ + x²eˣ = xeˣ(x + 2)
- y = sin x / x → dy/dx = (x cos x − sin x) / x² — y = sin x / x → dy/dx = (x cos x − sin x) / x²
a function is built from three or more layers, apply the chain rule repeatedly. For y = esin 2x, the derivative is esin 2x × cos 2x × 2 = 2 cos 2x · esin 2x.
当函数由三层以上构成时，需要反复使用链式法则。对于 y = esin 2x，导数为 esin 2x × cos 2x × 2 = 2 cos 2x · esin 2x。
5. Derivatives of Trigonometric Functions | 三角函数的导数
Trigonometric derivatives only hold in radians. This is a favourite exam trap: if a question states that angles are measured in degrees, the derivative of sin x is (π/180) cos x, not cos x. Always check the mode before differentiating.
三角函数的导数公式仅在弧度制下成立。这是考试中最爱的陷阱之一：若题目说明角度以度为单位，则 sin x 的导数是 (π/180) cos x，而不是 cos x。求导前务必确认角度制。
) = cos x d/dx (cos x) = −sin x d/dx (tan x) = 1 / cos²x = sec²x
in the cosine derivative. A useful way to remember the pair is that the derivative of the "co-" tangent result is proved by writing tan x = sin x / cos x and applying the quotient rule.
注意余弦导数中的符号变化。一个实用的记忆方法是：带“co-”的三角函数求导时总会出现负号。正切的结果可通过把 tan x 写成 sin x / cos x 再使用商法则来证明。
Combined with the chain rule, these give the standard results used constantly in IB questions:
结合链式法则，就得到IB题目中反复使用的标准结果：
- d/dx sin(kx) = k cos(kx), for example d/dx sin(3x) = 3 cos(3x) — d/dx sin(kx) = k cos(kx)，例如 d/dx sin(3x) = 3 cos(3x)
- d/dx cos(kx) = −k sin(kx) — d/dx cos(kx) = −k sin(kx)
- d/dx sin²x = 2 sin x cos x = sin 2x — d/dx sin²x = 2 sin x cos x = sin 2x
displacement x = A sin(ωt) has velocity v = Aω cos(ωt) and acceleration a = −Aω² sin(ωt).
在HL中，这些结果直接用于简谐运动的研究：位移 x = A sin(ωt) 的速度为 v = Aω cos(ωt)，加速度为 a = −Aω² sin(ωt)。
6. Exponential and Logarithmic Functions | 指数函数与对数函数
The exponential function eˣ is unique: it is its own derivative. This self-reproducing property is why e appears throughout growth and decay models in the AI course.
指数函数 eˣ 具有独一无二的性质：它的导数就是它自己。正是这种“自我复制”的特性，使 e 在AI课程的增长与衰减模型中无处不在。
d/dx (eˣ) = eˣ d/dx (eᵏˣ) = k eᵏˣ d/dx (ln x) = 1 / x (x > 0) d/dx (aˣ) = aˣ ln a
The natural logarithm appears as the inverse relationship: since ln x is the inverse of eˣ, its derivative 1/x completes the pair. Note that ln x is only defined for positive x, so any question involving it should state the domain.
自然对数与指数函数互为反函数：因为 ln x 是 eˣ 的反函数，所以它的导数 1/x 正好补全了这对关系。注意 ln x 只在 x 为正时有定义，因此涉及它的题目都应写明定义域。
- d/dx ln(3x) = 3 / (3x) = 1 / x, showing that the multiplicative constant cancels — d/dx ln(3x) = 3 / (3x) = 1 / x，可见乘法常数会被约掉
- d/dx e−2x = −2e−2x, the typical decay rate in half-life questions — d/dx e−2x = −2e−2x，这是半衰期问题中典型的衰减率
- d/dx x ln x = ln x + 1, by the product rule — d/dx x ln x = ln x + 1，由乘积法则得到
A model of the form P(t) = P₀ekt e rate of change is proportional to the current amount. Recognising this proportional relationship is a standard IB interpretation mark.
形如 P(t) = P₀ekt 的模型，其增长率为 dP/dt = kP，即变化率与当前数量成正比。识别出这一比例关系是IB常见的解释分点。
7. Higher-Order Derivatives | 高阶导数
Differentiating a derivative produces a second derivative, written f ″(x) or d²y/dx². Third and higher derivatives are denoted f ‴(x) and f (n)(x). Each successive derivative measures a further layer of change.
对导数再求导就得到二阶导数，记作 f ″(x) 或 d²y/dx²。三阶及更高阶导数记作 f ‴(x) 与 f (n)(x)。每多求一次导数，就多衡量一层变化。
f ″(x) = d/dx [f ′(x)] = d²y/dx²
In kinematics, if s(t) is displacement, then s′(t) is velocity and s″(t) is acceleration. In economics, if C(x) is total cost, then C′(x) is marginal cost and C″(x) indicates how the marginal cost itself is changing.
在运动学中，若 s(t) 表示位移，则 s′(t) 是速度，s″(t) 是加速度。在经济学中，若 C(x) 是总成本，则 C′(x) 是边际成本，C″(x) 表示边际成本自身的变化方式。
ndidate point of inflection. Setting y ′ = 0 gives x = 1 and x = 3, the stationary points.
一道例题：y = x³ − 6x² + 9x，则 y ′ = 3x² − 12x + 9，y ″ = 6x − 12。令 y ″ = 0 得 x = 2，即拐点的候选位置；令 y ′ = 0 得 x = 1 与 x = 3，即驻点。
8. Tangents, Normals and Rates of Change | 切线、法线与变化率
The tangent line at a point uses the derivative as its gradient. If the point of contact is (x₁, y₁) and the gradient there is m, the equation of the tangent is found from the point-slope form.
某点处的切线用导数作为其斜率。若切点为 (x₁, y₁)，该点斜率为 m，则切线方程由点斜式给出。
y − y₁ = m(x − x₁), where m = f ′(x₁)
er m ≠ 0. When m = 0 the tangent is horizontal and the normal is the vertical line x = x₁.
法线与切线垂直，因此只要 m ≠ 0，其斜率就是负倒数 −1/m。当 m = 0 时，切线水平，法线就是竖直线 x = x₁。
tated instant, or for the percentage rate of change, which is defined as (dy/dx) / y × 100%. A negative derivative simply means the quantity is decreasing at that moment.
变化率问题常要求给出某一时刻 dy/dx 的数值，或求百分比变化率，其定义为 (dy/dx) / y × 100%。导数为负只说明该量在那一刻正在减小。
Sign analysis of the derivative on either side of a point gives a global picture of where a function rises and falls, and this interval approach is often required before sketching a curve in Paper 2.
对某点两侧导数的符号分析，可以给出函数升降的整体图景；在Paper 2中作图之前，往往需要先做这种区间分析。
9. Stationary Points, Maxima and Minima | 驻点、极大值与极小值
. Stationary points are classified as local maxima, local minima, or points of inflection using either the first or the second derivative test.
当 f ′(x) = 0 时出现驻点，此处切线水平。可用一阶或二阶导数判别法把驻点分为局部极大值、局部极小值和拐点。
nd f ″(x) < 0 → local maximum f ′(x) = 0 and f ″(x) > 0 → local minimum
hat case fall back on the first derivative test: examine the sign of f ′(x) just to the left and just to the right of the stationary point. A change from positive to negative indicates a maximum; from negative to positive, a minimum; no sign change, a horizontal point of inflection.
二阶导数判别法更快，但当 f ″(x) = 0 时失效。此时应回到一阶导数判别法：检查驻点左、右两侧 f ′(x) 的符号。由正变负为极大值；由负变正为极小值；符号不变则为水平拐点。
t condition, exactly as f ′(x) = 0 is necessary but not sufficient for a stationary point.
拐点是凹凸性发生改变的位置，因此 f ″(x) 变号。注意这里的关键区别：f ″(x) = 0 是必要条件而非充分条件，正如 f ′(x) = 0 对驻点而言也只是必要条件。
as x → ±∞, then join the information smoothly. Every feature you can justify earns a mark.
Paper 2的作图要点：求截距，找出所有驻点与拐点，确定 x → ±∞ 时的趋势，再把信息平滑地连起来。每一个能说明理由的特征都可能得分。
10. Optimization and Kinematics Applications | 优化与运动学应用
Optimization problems ask for the largest or smallest possible value of a quantity subject to a constraint. The IB method is highly structured, and following it in order prevents most errors.
优化问题要求在给定约束下求某个量的最大或最小值。IB的解题方法非常
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