📚 A-Level WJEC Mathematics: Calculus Fundamentals Revision Guide | A-Level WJEC 数学：微积分基础 考点精讲
Calculus is the branch of mathematics that explores continuous change, split into two complementary halves: differentiation and integration. Differentiation gives us the instantaneous rate of change — the gradient of a curve at a point — while integration reverses the process to find the total accumulation of a quantity, such as the area under a curve. In the WJEC A-Level Mathematics specification, mastering these fundamentals is essential for paper 1 and beyond, as calculus appears in mechanics, optimisation and proof questions. This revision guide walks you through the core techniques, notation and exam strategies you need to build confidence and avoid common pitfalls.
微积分是研究连续变化的数学分支，分为相互对应的两大块：微分与积分。微分给出了瞬时的变化率——曲线在某一点的斜率；积分则是反转这一过程，求取量的累积，例如曲线下方的面积。在 WJEC A-Level 数学大纲中，掌握这些基础知识对试卷一以及后续内容至关重要，因为微积分还会出现在力学、最优化和证明题中。本考点精讲将带你梳理核心技巧、符号表示与应试策略，帮助建立自信，避开常见失分点。
1. Definition of Derivative and Notation | 导数的定义与记号
The derivative of a function f(x) is defined as the limit, as h tends to 0, of the difference quotient: f'(x) = lim (h → 0) [f(x + h) – f(x)] / h. It measures the gradient of the tangent to the curve y = f(x) at any point. You may encounter alternative notations: dy/dx (Leibniz), f'(x) (Lagrange), or even D_xy on WJEC papers. Each conveys the same concept — the instantaneous rate of change.
函数 f(x) 的导数定义为差商在 h 趋于 0 时的极限：f'(x) = lim (h→0) [f(x+h) – f(x)] / h。它衡量曲线 y=f(x) 在任一点处切线的斜率。你会在 WJEC 试卷中见到不同的记号：dy/dx（莱布尼茨记号）、f'(x)（拉格朗日记号），甚至 D_xy。它们表达的其实是同一个概念——瞬时变化率。
A function is differentiable at a point if this limit exists. For WJEC, you only need to apply differentiation rules, not compute limits from first principles unless explicitly asked. However, understanding the definition helps you explain why the derivative of a constant is zero and why power rule works.
如果该极限存在，函数在该点可导。对于 WJEC 考试，除非明确要求从第一性原理推导，否则你只需要运用求导法则。但理解定义有助于解释为什么常数的导数为零，以及幂法则为什么成立。
2. Power Rule and Basic Differentiation Rules | 幂法则与基本求导规则
For a term of the form axⁿ, differentiate by multiplying the coefficient by the power and reducing the power by one: d/dx (axⁿ) = a·n·xⁿ⁻¹. This rule holds for any real constant n, including negative and fractional powers. For instance, d/dx (x⁴) = 4x³, and d/dx (√x) = d/dx (x^(1/2)) = (1/2)x^(-1/2). Always rewrite roots and fractions before differentiating.
对于形如 axⁿ 的项，求导时用系数乘以幂指数，再将幂指数减一：d/dx (axⁿ) = a·n·xⁿ⁻¹。这一规则适用于任意实数常数 n，包括负数和分数指数。例如，d/dx (x⁴) = 4x³，而 d/dx (√x) = d/dx (x^(1/2)) = (1/2)x^(-1/2)。一定要先将根式和分母形式改写为幂的形式再求导。
The constant multiple rule states that d/dx [c·f(x)] = c·f'(x), and the sum/difference rule says d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Apply these to polynomials term by term. Example: differentiate f(x) = 3x⁵ – 2x³ + 7x – 4. The derivative is f'(x) = 15x⁴ – 6x² + 7, since the constant 4 differentiates to 0.
常数倍法则：d/dx [c·f(x)] = c·f'(x)；和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。对多项式可以逐项使用。示例：对 f(x) = 3x⁵ – 2x³ + 7x – 4 求导，得到 f'(x) = 15x⁴ – 6x² + 7，因为常数 4 的导数为 0。
3. Derivatives of Exponential and Natural Logarithm Functions | 指数函数与自然对数函数的导数
The exponential function eˣ is unique because its derivative is itself: d/dx (eˣ) = eˣ. When the exponent is a linear function of x, such as e^(kx), the chain rule gives d/dx (e^(kx)) = k e^(kx). For instance, d/dx (e^(2x)) = 2e^(2x). In WJEC questions, e is always the base of natural logarithm, so you never need to convert to other bases.
指数函数 eˣ 的独特之处在于它的导数是它本身：d/dx (eˣ) = eˣ。当指数是 x 的线性函数，如 e^(kx)，链式法则给出 d/dx (e^(kx)) = k e^(kx)。例如，d/dx (e^(2x)) = 2e^(2x)。在 WJEC 试题中，e 永远是自然对数的底数，你无需转换其他底数。
The natural logarithm function y = ln x has derivative d/dx (ln x) = 1/x, defined for x > 0. For compound arguments like ln(kx), the chain rule gives d/dx (ln(kx)) = 1/x again, because the constant k cancels out. For ln(3x), derivative is 1/x. However, be careful: d/dx (ln(3x+2)) uses the chain rule: 3/(3x+2).
自然对数函数 y = ln x 的导数为 d/dx (ln x) = 1/x，定义域为 x>0。对于复合自变量如 ln(kx)，链式法则再次给出 1/x，因为常数 k 会被抵消。对于 ln(3x)，导数仍然是 1/x。但要注意：d/dx (ln(3x+2)) 需要链式法则，结果为 3/(3x+2)。
4. Derivatives of Trigonometric Functions | 三角函数的导数
WJEC AS expects you to know the derivatives of sin x and cos x. d/dx (sin x) = cos x, and d/dx (cos x) = -sin x. These results require x to be in radians — differentiation formulas only hold when angles are measured in radians, not degrees. Always convert degrees to radians if a problem gives an angle in degrees for a derivative question.
WJEC AS 阶段需要掌握 sin x 与 cos x 的导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x。这些结果仅在角度以弧度为单位时成立——微分公式只适用于弧度制，而非角度制。如果在求导题目中遇到度数，务必先转换为弧度。
When the argument includes a constant multiplier, apply the chain rule. For example, d/dx [sin(2x)] = 2 cos(2x), and d/dx [cos(5x)] = -5 sin(5x). You may also need to differentiate tan x = sin x / cos x using the quotient rule, which gives sec² x, but this is often covered later. Keep these basic sinusoids and their radian measures firmly in mind.
当自变量含有常数倍数时，使用链式法则。例如 d/dx [sin(2x)] = 2 cos(2x)，d/dx [cos(5x)] = -5 sin(5x)。你可能还需要用商法则求 tan x = sin x/cos x 的导数，得到 sec² x，不过这往往放在后面。牢记这些基本正弦、余弦函数以及弧度制要求。
5. Equations of Tangents and Normals | 切线方程与法线方程
To find the equation of a tangent to a curve y = f(x) at a point (a, f(a)), first compute the derivative f'(a) to obtain the gradient m_t. Then use the point-gradient form: y – f(a) = m_t (x – a). For the normal, which is perpendicular to the tangent, the gradient m_n satisfies m_t × m_n = -1, so m_n = -1/m_t (provided m_t ≠ 0). Then plug into the same point-gradient form.
求曲线 y=f(x) 在点 (a, f(a)) 的切线方程，首先计算导数 f'(a) 得到斜率 m_t，然后利用点斜式：y – f(a) = m_t (x – a)。法线垂直于切线，其斜率 m_n 满足 m_t × m_n = -1，即 m_n = -1/m_t（前提是 m_t 不为零）。同样代入点斜式即可。
WJEC exam questions often ask for both equations or expect you to find where the tangent is horizontal (gradient 0) or parallel to a given line. Remember: horizontal tangents occur when f'(x) = 0; parallel to line y = mx + c means f'(x) = m. Always write the final answer in the requested form, e.g., ax + by + c = 0.
WJEC 考题经常要求同时写出切线方程和法线方程，或让你找到切线水平（斜率为零）或与给定直线平行的点。记住：水平切线发生在 f'(x)=0；与直线 y=mx+c 平行意味着 f'(x)=m。最后答案一定要写成题目要求的格式，比如 ax+by+c=0。
6. Increasing and Decreasing Functions | 函数的递增与递减区间
A function f(x) is increasing on an interval if f'(x) > 0 for all x in that interval, and decreasing if f'(x) < 0. To determine intervals, find the derivative, solve f'(x) = 0 to locate boundary points (stationary points), and test sign of f'(x) on either side. Use a sign diagram or a simple table to organise the signs.
如果在某区间内 f'(x)>0，函数在该区间递增；如果 f'(x)<0，函数递减。要找出递增递减区间，先求导，解方程 f'(x)=0 找到分界点（驻点），再检测两侧 f'(x) 的符号。可以使用符号表或简易表格整理符号变化。
For example, given f(x) = x³ – 3x, then f'(x) = 3x² – 3 = 3(x-1)(x+1). Setting f'(x)=0 gives x = 1, -1. Test x < -1, f'(x) > 0 (increasing); between -1 and 1, f'(x) < 0 (decreasing); x > 1, f'(x) > 0 (increasing). State intervals as: increasing for x < -1 and x > 1; decreasing for -1 < x < 1.
例如，给定 f(x)=x³-3x，则 f'(x)=3x²-3=3(x-1)(x+1)。令 f'(x)=0 得 x=1，-1。代入检测：x<-1 时 f'(x)>0（递增）；-1 Stationary points occur where f'(x) = 0. To classify them as maximum, minimum or point of inflection, the second derivative f”(x) is used. If f”(x) < 0 at the point, it's a local maximum (concave down). If f''(x) > 0, it’s a local minimum (concave up). If f”(x) = 0, the test is inconclusive and you must examine the sign change of f'(x) on either side. 驻点出现在 f'(x)=0 处。要区分极大值、极小值或拐点，可借助二阶导数 f”(x)。若在该点 f”(x)<0，则为局部极大值（凹向下）；若 f''(x)>0，则为局部极小值（凹向上）；若 f”(x)=0，则二阶导数判别法失效，需要检查 f'(x) 在两侧的符号变化。 Using the previous example f(x)=x³-3x: f'(x)=3x²-3, f”(x)=6x. At x=1, f”(1)=6>0 → local minimum; at x=-1, f”(-1)=-6<0 → local maximum. Always substitute x back into original f(x) to state coordinates, e.g., (1, -2) is a minimum and (-1, 2) is a maximum. WJEC expects ordered pairs and classification labels. 继续用上例 f(x)=x³-3x：f'(x)=3x²-3，f”(x)=6x。在 x=1 处，f”(1)=6>0 → 局部极小值；在 x=-1 处，f”(-1)=-6<0 → 局部极大值。务必代回原函数求出坐标，如 (1, -2) 为极小值点，(-1, 2) 为极大值点。WJEC 要求给出有序数对并标明类别。 Indefinite integration reverses differentiation. To integrate a power of x: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ -1. The constant of integration C is essential because an indefinite integral represents a family of functions whose derivative is the integrand. Always add + C in WJEC indefinite integral answers unless the question specifically asks for a particular antiderivative. 不定积分是对微分的逆运算。对 x 的幂次积分：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n≠-1。积分常数 C 必不可少，因为不定积分表示一族导数等于被积函数的函数。在 WJEC 的不定积分答案中必须始终加上 +C，除非题目明确要求某个特定的原函数。 For sums and constant multiples: ∫ k·f(x) dx = k ∫ f(x) dx, and ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx. Before integrating, simplify the expression: split fractions, rewrite roots as fractional powers. Example: ∫ (3x² + 2/√x) dx = ∫ (3x² + 2x^(-1/2)) dx = x³ + 4x^(1/2) + C. 对和与常数倍：∫ k·f(x) dx = k ∫ f(x) dx，∫ [f(x)±g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。积分前先简化表达式：拆分分式，将根式化为分数幂。例如：∫ (3x² + 2/√x) dx = ∫ (3x² + 2x^(-1/2)) dx = x³ + 4x^(1/2) + C。 For special functions: ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C; ∫ cos x dx = sin x + C; ∫ sin x dx = -cos x + C. These must be memorised for the WJEC exam as they are the inverses of the respective derivatives. 特殊函数的积分：∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C；∫ cos x dx = sin x + C；∫ sin x dx = -cos x + C。这些必须牢记，因为它们是相应导数的逆运算，WJEC 考试中经常直接考查。 A definite integral ∫ₐᵇ f(x) dx computes the net area between the curve and the x-axis from x=a to x=b. The Fundamental Theorem of Calculus states that if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) – F(a). First find the indefinite integral, but do not include +C, then substitute the upper and lower limits and subtract. 定积分 ∫ₐᵇ f(x) dx 计算的是曲线与 x 轴之间从 x=a 到 x=b 的净面积。微积分基本定理指出，如果 F'(x)=f(x)，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。先求出不定积分（此时不加常数 C），然后代入上限和下限并相减。 Example: Evaluate ∫₁³ (2x+1) dx. Antiderivative: x² + x. Then compute (3²+3) – (1²+1) = (9+3)-(1+1)=10. So the definite integral equals 10. Always show clear substitution steps; WJEC awards marks for correct evaluation of F(b) – F(a). 示例：计算 ∫₁³ (2x+1) dx。原函数为 x² + x，再计算 (3²+3)-(1²+1) = (9+3)-(1+1)=10，故积分值为 10。始终清晰地写出代入步骤；WJEC 评分重视 F(b)-F(a) 的正确运算过程。 The area under y = f(x) between x=a and x=b is given by ∫ₐᵇ |f(x)| dx if the curve crosses the x-axis. When f(x) is negative, the integral gives a negative value; to find the true geometric area, split the interval at the roots and add absolute values of each section. Always draw a quick sketch in the exam to see which regions lie above or below the x-axis. 若曲线在 x=a 与 x=b 之间穿过 x 轴，其下方几何面积为 ∫ₐᵇ |f(x)| dx。当 f(x) 为负值时，积分结果为负；要得到真实的几何面积，须在零点处分割区间，并将各段取绝对值相加。考试中务必快速画图，确定哪些区域位于 x 轴上方还是下方。 For area between two curves y = f(x) and y = g(x) from x=a to x=b, the enclosed area is ∫ₐᵇ [upper curve – lower curve] dx. Determine which function is on top over the interval. If they cross, split into multiple integrals. WJEC may ask for area bounded by curves and lines, e.g., find area between y = x² and y = 4. Solve intersections x = -2, 2, then area = ∫₋₂² (4 – x²) dx. 求两条曲线 y=f(x) 与 y=g(x) 在 x=a 到 x=b 之间的面积时，包围面积为 ∫ₐᵇ [上方曲线 – 下方曲线] dx。需要先判断在该区间内哪条函数在上方。如果有交叉，则拆分成多个积分区间。WJEC 可能会问曲线与直线所围面积，例如求 y=x² 与 y=4 之间的面积：解交点 x=-2, 2，面积 = ∫₋₂² (4 – x²) dx。 One of the most frequent errors is forgetting the constant of integration +C in indefinite integrals — this costs an accuracy mark every time. Another is misapplying the power rule for integration when n = -1, which leads to ln|x|, not division by zero. Also, be careful with brackets when substituting negative limits; write them step by step to avoid sign errors. 最常见的错误之一是在不定积分中漏掉积分常数 +C——每次都会丢一个准确分。另一个是 n=-1 时错误套用幂法则积分，正确结果应为 ln|x|，而不是除以零。代入负下限时也要注意括号，逐步书写以避免符号错误。 In differentiation, candidates often confuse the derivative of sin x and cos x — remember, derivative of sin is cos, derivative of cos is -sin. Additionally, when finding stationary points, always confirm classification with either first or second derivative test; simply stating the point without justification loses marks. On WJEC papers, show all working clearly: the method marks are generous when steps are logical. 在微分中，考生常常混淆 sin x 与 cos x 的导数——记住，sin 的导数是 cos，cos 的导数是 -sin。此外，找驻点时，一定要用一阶导数或二阶导数判别法确认极大极小；只说是什么点而不说明理由会丢分。在 WJEC 试卷上，务必清晰展示所有解题步骤：只要逻辑步骤清晰，方法分往往给得很宽松。 Finally, practice converting all roots and rational expressions to index form before differentiating or integrating. Keep a summary table of basic derivatives and integrals handy during revision. Time management is crucial; if an integral looks messy, simplify algebraically first — often the question is designed to test that simplification skill. 最后，在微分或积分前，务必将所有根式和有理式转化为指数形式。复习时随身携带基本导数与积分对照表。时间管理也很关键；如果积分式子看起来很乱，先进行代数化简——题目往往就是故意考查你的化简能力。 Published by TutorHao | Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
7. Stationary Points and the Second Derivative Test | 驻点与二阶导数判别法
8. Indefinite Integration and the Reverse Power Rule | 不定积分与幂法则逆运算
9. Definite Integration and the Fundamental Theorem | 定积分与微积分基本定理
10. Area Under a Curve and Between Curves | 曲线下方面积与曲线间面积
11. Common Mistakes and Examination Tips | 常见错误与应试技巧
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