📚 Differential Calculus: IB WJEC Exam Guide | IB WJEC 数学：微分 考点精讲
Differential calculus is a cornerstone of the IB and WJEC Mathematics syllabus, bridging the gap between algebraic manipulation and real-world change. Mastering derivatives not only unlocks higher marks in exams but also lays the groundwork for topics such as kinematics, optimisation, and integration. This guide walks you through every key concept you need, from first principles to advanced applications, with a clear focus on exam-style reasoning.
微分是 IB 与 WJEC 数学课程的重要基石，它把代数运算与现实世界的变化联系起来。掌握导数不仅能帮助你在考试中取得高分，也为运动学、最优化和积分等后续内容打下坚实基础。本文将从第一性原理到高级应用，带你逐一攻克关键考点，并紧扣考试中常见的推理方式。
1. The Definition of the Derivative | 导数的定义
a is defined as the limit of the average rate of change as the interval approaches zero: f'(a) = limₕ→₀ [f(a+h) – nd geometrically it gives the gradient of the tangent to the curve y = f(x) at that point.
函数 f(x) 在 x = a 处的导数定义为当区间趋近于零时平均变化率的极限：f'(a) = limₕ→₀ [f(a+h) – f(a)] / h。若该极限存在，它就表示 f 关于 x 的瞬时变化率，几何意义是曲线 y = f(x) 在该点处切线的斜率。
eed, set up the difference quotient, simplify the numerator carefully, and then let h → 0. Remember that the limit must yield a finite value; otherwise the derivative does not exist at that point.
在 IB 与 WJEC 考试中，你可能会被要求用第一性原理求导，通常针对 f(x) = x² 或 f(x) = 1/x 等简单函数。要得分，需要正确写出差商表达式，仔细化简分子，再令 h → 0。注意极限必须为有限值，否则函数在该点不可导。
The notation f'(x), dy/dx, or d/dx [f(x)] are all used interchangeably. The Leibniz notation dy/dx is particularly useful for emphasising the variable with respect to which we differentiate and for understanding the chain rule.
导数符号 f'(x)、dy/dx 和 d/dx[f(x)] 可以互换使用。莱布尼茨记号 dy/dx 尤其适合强调对哪个变量求导，也有助于理解链式法则。
2. Basic Differentiation Rules | 基本求导法则
The power rule states that for f(x) = xⁿ, where n is any real constant, f' rm xⁿ. For constant functions f(x) = k, the derivative is zero because there is no change.
幂函数求导法则指出，对于 f(x) = xⁿ（n 为任意实常数），f'(x) = n xⁿ⁻¹。只要能把函数表示成 xⁿ 的形式，这一法则就能处理大量多项式与有理函数。常函数 f(x) = k 的导数为零，因为函数值不变。
e and the sum rule allow us to differentiate linear combinations. If f(x) = c g(x), then f'(x) = c g'(x). If f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x). These rules mean you can differentiate a polynomial term by term.
常数倍法则与和差法则使我们能对线性组合求导。若 f(x) = c g(x)，则 f'(x) = c g'(x)；若 f(x) = g(x) ± h(x)，则 f'(x) = g'(x) ± h'(x)。利用这些法则，你就可以逐项对多项式求导。
Table of essential basic derivatives:
| f(x) | f'(x) |
|---|---|
| k (constant) | 0 |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| aˣ (a > 0) | aˣ ln a |
基本导数公式表：
| f(x) | f'(x) |
|---|---|
| k (常数) | 0 |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| aˣ (a > 0) | aˣ ln a |
3. Chain Rule: Compositions of Functions | 链式法则：复合函数求导
ates that dy/dx = dy/du × du/dx, where u = g(x) and y = f(u). In prime notation, [f(g(x))]' = f'(g(x)) g' ssential whenever you see a function inside another function, such as (3x²+1)⁵ or sin(2x).
链式法则用于对复合函数 f(g(x)) 求导。其公式为 dy/dx = dy/du × du/dx，其中 u = g(x)，y = f(u)。使用一撇记号可写成 [f(g(x))]' = f'(g(x)) g'(x)。每当你遇到“函数套函数”的形式，如 (3x²+1)⁵ 或 sin(2x)，都需要用到该法则。
A common exam technique is to identify the 'outer' and 'inner' functions, differentiate the outer while keeping the inner unchanged, then multiply by the derivative of the inner. For example, to differentiate y = √x²+1, rewrite as y = (x²+1)1/2. The outer function is u1/2, inner is u = x²+1. Derivative: (1/2)(x²+1)–1/2 × 2x = x / √x²+1.
考试中常用的技巧是分清“外层”与“内层”函数，外层求导时保持内层不变，再乘以内层导数。例如，对 y = √x²+1 求导，可改写为 y = (x²+1)1/2。外层函数为 u1/2，内层为 u = x²+1。求导得：(1/2)(x²+1)–1/2 × 2x = x / √x²+1。
Be careful with repeated application: some functions require two or more layers of the chain rule, such as sin³(4x). Work from the outermost function inward, multiplying the derivatives as you go.
需要留意多次使用链式法则的情况：有些函数包含两层以上的复合，如 sin³(4x)。求解时应从最外层开始逐步向内求导，并将各层导数相乘。
4. Product and Quotient Rules | 乘积法则与商法则
o differentiable functions u(x) and v(x) are multiplied, the product rule states that (uv)' = u' v + u v'. This rule must be memorised accurately—many students mistakenly write u' v' or forget the second term.
当两个可导函数 u(x) 与 v(x) 相乘时，乘积法则要求 (uv)' = u' v + u v'。必须准确记住该公式——许多考生会误写成 u' v' 或漏掉第二项。
For division, the quotient rule gives (u/v)' = (u' v – u v') / v². The order of terms in the numerator matters: it is always 'bottom times derivative of the top minus top times derivative of the bottom' over the square of the bottom. In Leibniz notation, if y = u/v, then dy/dx = (v du/dx – u dv/dx) / v².
对于相除的情形，商法则给出 (u/v)' = (u' v – u v') / v²。分子的相减顺序至关重要：始终是“分母乘以分子的导数 减去 分子乘以分母的导数”，再除以分母的平方。用莱布尼茨记号可写为：若 y = u/v，则 dy/dx = (v du/dx – u dv/dx) / v²。
⁻¹ and differentiated using the product rule. This approach can sometimes reduce algebraic errors.
很多时候你可以通过将分式改写成负指数形式的乘积来避免使用商法则。例如，y = x/(x+1) 可写为 x (x+1)⁻¹，再用乘积法则求导。这一方法有时能减少代数运算中的错误。
| Rule | Formula |
|---|---|
| Product (乘积) | d/dx [u v] = u' v + u v' |
| Quotient (商) | d/dx [u/v] = (u' v – u v') / v² |
5. Derivatives of Trigonometric Functions | 三角函数求导
The derivatives of sine and cosine are cyclic: d/dx sin x = cos x, d/dx cos x = – ntly ask for the derivative of sin(kx) or cos(kx) using the chain rule, giving k cos kx and –k sin kx respectively.
正弦和余弦的导数互为循环：d/dx sin x = cos x，d/dx cos x = – sin x。由此可推得 tan x 的导数为 sec² x，这可以通过对 sin x / cos x 使用商法则得出，或直接记忆为 1/cos² x。WJEC 与 IB 常考形如 sin(kx) 或 cos(kx) 的求导，使用链式法则分别得到 k cos kx 与 –k sin kx。
csc x, and cot x, learn their standard derivatives: d/dx sec x = sec x tan x, d/dx csc x = –csc x cot x, d/dx cot x = – ntiation questions but can appear in more advanced problems or integration by parts later.
对于 sec x、csc x 和 cot x 等倒数三角函数，需要掌握它们的标准导数公式：d/dx sec x = sec x tan x，d/dx csc x = –csc x cot x，d/dx cot x = –csc² x。它们在基础求导题中较少出现，但可能在较复杂的问题或后续分部积分中遇到。
in rule. The final derivative is 2 sin(3x) × cos(3x) × 3 = 6 sin(3x) cos(3x) = 3 sin(6x) (using the double-angle identity). This illustrates how multiple rules combine.
求导形如 sin²(3x) 的表达式时，应将其视为 (sin(3x))²。外层函数为 u²，内层为 sin(3x)，而 sin(3x) 本身仍需使用链式法则。最终导数为 2 sin(3x) × cos(3x) × 3 = 6 sin(3x) cos(3x) = 3 sin(6x)（利用倍角公式化简）。此例展示了多种法则的综合运用。
6. Exponential and Logarithmic Derivatives | 指数函数与对数函数求导
The natural exponential function eˣ is its own derivative, d/dx eˣ = eˣ, making it unique. General exponential functions aˣ have derivative aˣ a. When the exponent is a function of x, apply the chain rule: d/dx e^(g(x)) = e^(g(x)) g'(x). This is one of the most commonly tested patterns in both IB and WJEC papers.
自然指数函数 eˣ 的导数等于它本身，即 d/dx eˣ = eˣ，这使得它极为特殊。一般指数函数 aˣ 的导数为 aˣ ln a。当指数为 x 的函数时，需使用链式法则：d/dx e^(g(x)) = e^(g(x)) g'(x)。这是 IB 与 WJEC 试卷中最常见的考查点之一。
For logarithms, d/dx ln x = 1/x for x > 0. For ln(g(x)), the chain rule gives g'(x)/g(x). The derivative of logₐ x is 1/(x ln a). You might also see l n |x| ve values; the derivative is still 1/x, but this is generally met in integration contexts.
对数函数的导数：d/dx ln x = 1/x（x > 0）。对于 ln(g(x))，使用链式法则得 g'(x)/g(x)。logₐ x 的导数为 1/(x ln a)。你也可能遇到 ln |x| 的形式（x 为负值时），其导数仍为 1/x，不过这主要在积分中用到。
licated products, quotients, or functions with variable bases and exponents, such as y = xˣ Take natural logs of both sides, ln y = x ln x, then differentiate implicitly to obtain y' = y (ln x + 1) = xˣ (ln x + 1). This technique is highly valued in exam solutions.
常见的巧妙方法是利用对数求导法处理复杂的乘积、商或底数与指数均为变量的函数，如 y = xˣ。对两边取自然对数：ln y = x ln x，然后隐函数求导得到 y' = y (ln x + 1) = xˣ (ln x + 1)。这一技巧在考试答案中十分受青睐。
7. Implicit Differentiation | 隐函数求导
lying the chain rule to terms containing y. For example, given x² + y² = 1, differentiating gives 2x + 2y dy/dx = 0, so dy/dx = –x/y.
当 y 未明确表示为 x 的函数时，我们需要对等式两边关于 x 求导，将 y 视作 x 的函数，并对含 y 的项使用链式法则。例如，对于 x² + y² = 1，求导得 2x + 2y dy/dx = 0，从而 dy/dx = –x/y。
tes) or for finding tangents to ellipses and hyperbolas given in general form. It also appears in related rates problems, where two or more variables change with time.
对于一些由方程定义的曲线，如 x³ + y³ = 6xy（笛卡尔叶形线），或求一般形式的椭圆、双曲线的切线时，隐函数求导是必不可少的方法。它同样出现在相关变化率问题中，即两个或多个变量随时间变化的情形。
dy/dx. Check that your final expression for dy/dx is in terms of x and y only; then you can substitute point coordinates to find gradients.
注意在对含有 y 的乘积项求导时要格外小心。对于 xy² 这样的项，必须使用乘积法则关于 x 求导：d/dx [x y²] = y² + 2xy dy/dx。务必确认最终得到的 dy/dx 表达式只含 x 和 y，然后代入驻点坐标即可求得斜率。
8. Higher-Order Derivatives | 高阶导数
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 is used to determine the concavity of a function and to classify stationary points (maxima, minima, points of inflection).
二阶导数，记作 f''(x) 或 d²y/dx²，是一阶导数的导数。它衡量斜率的变化率，用于判断函数的凹凸性，并用于判别驻点类型（极大值、极小值、拐点）。
d WJEC set questions where you must find and interpret second derivatives in motion contexts, such as determining when a particle is accelerating or decelerating.
在运动学中，若位移 s = f(t)，则速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。IB 和 WJEC 都会设置要求你求解并解释二阶导数含义的题目，例如判断粒子何时在加速或减速。
in rules. For trigonometric functions, derivatives cycle every four steps: sin x → cos x → –sin x → – . Identifying such patterns can save time when required to find, say, the 10th derivative.
求更高阶导数常常需要反复使用乘积、商与链式法则。对于三角函数，导数每四步循环一次：sin x → cos x → –sin x → –cos x → sin x。发现这种规律能在要求“求第 10 阶导数”时节省大量时间。
9. Application: Tangent and Normal Lines | 应用：切线方程与法线
The gradient of the tangent to the curve y = f(x) at x = a is f'(a). The equation of the tangent is y – f(a) = f'(a)(x – The normal is perpendicular to the tangent, so its gradient is the negative reciprocal: mN = –1 / f'(a), provided f'(a) ≠ 0. The normal equation is y – f(a) = (–1/f'(a))(x – a).
曲线 y = f(x) 在 x = a 处的切线斜率为 f'(a)。切线方程为 y – f(a) = f'(a)(x – a)。法线与切线垂直，因此其斜率为负倒数：mN = –1 / f'(a)（假设 f'(a) ≠ 0）。法线方程为 y – f(a) = (–1/f'(a))(x – a)。
dy/dx = 0) or vertical (dx/dy = 0). The latter arises in implicit differentiation when the denominator of dy/dx is zero. Always state the equation in the required form, typically ax + by + c = 0.
考试题可能给出一条曲线方程，要求你求某点处的切线方程，或求切线为水平的位置（dy/dx = 0）以及切线为竖直的位置（dx/dy = 0）。后者出现在隐函数求导中，当 dy/dx 的分母为零时成立。最后务必按要求形式写出方程，通常为 ax + by + c = 0。
Check your final answer by substituting the original point back into the line equation. A small slip in differentiation can lead to an incorrect gradient, which can cost several marks if the rest of the working follows from that error.
求出方程后，把原点的坐标代入检验。微分过程中的一个小笔误可能导致斜率出错，若后续推理皆基于此错误，可能损失大量分数。
10. Application: Rates of Change and Optimisation | 应用：变化率与最优化
r/dt (chain rule). Exam questions typically require expressing one rate of change in terms of others, often using geometric relationships from circles, spheres, or cones.
相关变化率问题涉及随时间 t 变化的变量。若 V 表示体积，r 表示半径，且二者均依赖于 t，则 dV/dt = dV/dr × dr/dt（链式法则）。考试题通常要求用其他变化率表示某一变化率，常借助圆、球或圆锥的几何关系。
mum or minimum values of a function on a given interval. Steps: (1) Express the quantity to be optimised as a function of one variable. (2) Differentiate and find stationary points (f' e test or a sign diagram to classify. (4) Check endpoints if the domain is closed. (5) Write the conclusion with units.
最优化问题寻找函数在给定区间上的最大值或最小值。解题步骤：(1) 用单一变量表示待优化量；(2) 求导并令 f'(x) = 0 找出驻点；(3) 使用二阶导数判别法或符号表判断极值类型；(4) 若定义域为闭区间，检查端点值；(5) 写出带单位的结论。
A classic problem: 'Find the dimensions of a cylindrical can of minimum surface area for a fixed volume.' Express surface area in terms of radius r, differentiate, set derivative to zero, and verify it is a minimum. IB and WJEC examiners expect clear algebraic working and justification of the nature of the stationary point.
经典问题如：“在体积固定的情况下，求圆柱形罐子的最小表面积”。用半径 r 表示表面积，求导并置零，再验证其为极小值。IB 和 WJEC 的阅卷人期望看到清晰的代数推导以及驻点性质的说明。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most frequent errors is forgetting the chain rule when differentiating composite functions, especially inside trigonometric or exponential expressions. Always ask: 'Is there an inner function?' before you start. A simple checklist: identify inner and outer, differentiate outer, multiply by derivative of inner.
最常见的错误之一是在对复合函数求导时忘记使用链式法则，尤其是在三角函数或指数函数内部。开始求导前先问自己：“有内层函数吗？”简单的检查流程：确定内外层，对外层求导，再乘以内层的导数。
Another common mistake is mismanaging signs. The derivative of cos x is negative, and the quotient rule has a minus sign in the numerator—swapping the terms leads to an error. Write the rules on a formula card and refer to it until they become automatic. Pay special attention to the derivative of e⁻ˣ (which is –e⁻ˣ) and ln(–x) (still 1/x).
另一个常见错误是符号处理不当。cos x 的导数为负值，商法则的分子中有减号——一旦交换项的顺序就会出错。把公式写在小卡片上经常查阅，直至熟练为止。要特别注意 e⁻ˣ 的导数为 –e⁻ˣ，而 ln(–x) 的导数仍然是 1/x。
torise, expand, or rewrite using exponent laws where possible. For instance, y = (x²+1)(x³– before differentiating, avoiding the product rule unless the question instructs otherwise.
求导前不化简会使问题变得困难许多。应尽可能先因式分解、展开或用指数法则重写表达式。例如，y = (x²+1)(x³–2) 可先乘开再求导，这样就不必使用乘积法则，除非题目明确要求使用该法则。
Finally, always interpret the context. If a question asks for the rate at which the water level is falling, the answer should be negative. Similarly, for optimisation, ensure you give the required dimensions, not just the value of the variable. Units are essential—omitting them can cost marks in both boards.
最后，务必要结合题目背景解释答案。如果题目问的是“水位下降的速率”，答案应为负值。同样，在最优化问题中，要给出所要求的尺寸，而不仅仅是变量的数值。单位至关重要——忽略单位在两大考试局中都会导致扣分。
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