A-Level数学 微积分 微分 积分 求导 技巧
2| 2| 3| 3|Overview: Edexcel A-Level Calculus 概览
4| 4| 5| 5|Calculus is the beating heart of A-Level Mathematics. In the Edexcel specification, it spans Pure Mathematics 1 through 4, appearing in roughly 40% of all exam questions. Whether you are sitting the AS or the full A-Level, mastery of differentiation and integration is non-negotiable.
6| 6| 7| 7|微积分是 A-Level 数学的灵魂。在 Edexcel 考试大纲中，微积分横跨 Pure Mathematics 1 到 4 四本书，出现在大约 40% 的考题中。无论你参加 AS 还是完整 A-Level，掌握微分和积分是必须的。
8| 8| 9| 9|The syllabus splits neatly into two halves: differentiation (finding gradients, rates of change, and turning points) and integration (finding areas, volumes, and reversing differentiation). Both connect through the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations.
10| 10| 11| 11|考试大纲分为两半：微分（求梯度、变化率和拐点）和积分（求面积、体积以及微分的逆运算）。两者通过微积分基本定理连接::微分和积分互为逆运算。
12| 12| 13| 13|Differentiation: The Rules 微分法则
14| 14| 15| 15|Edexcel expects fluency in five core differentiation rules. Start with the power rule: if y = x^n, then dy/dx = n x^(n-1). This applies to polynomials of any degree. For example, if y = 3x^4 - 2x^3 + 5x - 7, then dy/dx = 12x^3 - 6x^2 + 5.
16| 16| 17| 17|Edexcel 要求熟练掌握五个核心微分法则。从幂法则开始：如果 y = x^n，那么 dy/dx = n x^(n-1)。这适用于任何次数的多项式。例如，如果 y = 3x^4 - 2x^3 + 5x - 7，那么 dy/dx = 12x^3 - 6x^2 + 5。
18| 18| 19| 19|The product rule handles functions multiplied together: if y = u(x) v(x), then dy/dx = u dv/dx + v du/dx. A classic exam question asks you to differentiate y = x^2 sin x. Here u = x^2 (du/dx = 2x) and v = sin x (dv/dx = cos x), giving dy/dx = x^2 cos x + 2x sin x.
20| 20| 21| 21|乘法法则处理两个函数相乘的情况：如果 y = u(x) v(x)，那么 dy/dx = u dv/dx + v du/dx。经典考题：求 y = x^2 sin x 的导数。这里 u = x^2 (du/dx = 2x)，v = sin x (dv/dx = cos x)，得到 dy/dx = x^2 cos x + 2x sin x。
22| 22| 23| 23|The quotient rule tackles fractions: if y = u/v, then dy/dx = (v du/dx - u dv/dx) / v^2. Remember: "low d-high minus high d-low, over low squared." For y = (2x+1)/(x-3), differentiate numerator (du/dx=2) and denominator (dv/dx=1), giving dy/dx = [(x-3)(2) - (2x+1)(1)] / (x-3)^2 = (2x-6-2x-1)/(x-3)^2 = -7/(x-3)^2.
24| 24| 25| 25|除法法则处理分式：如果 y = u/v，那么 dy/dx = (v du/dx - u dv/dx) / v^2。记住口诀：分母乘分子的导数，减去分子乘分母的导数，再除以分母的平方。对于 y = (2x+1)/(x-3)，分子导数 du/dx=2，分母导数 dv/dx=1，得到 dy/dx = -7/(x-3)^2。
26| 26| 27| 27|The chain rule is arguably the most important single technique in A-Level calculus. When y is a function of u, and u is a function of x: dy/dx = (dy/du) × (du/dx). Differentiate y = (3x^2 + 2)^5. Let u = 3x^2 + 2, so y = u^5. Then dy/du = 5u^4 and du/dx = 6x. So dy/dx = 5(3x^2+2)^4 × 6x = 30x(3x^2+2)^4.
28| 28| 29| 29|链式法则可以说是 A-Level 微积分中最重要的单一技巧。当 y 是 u 的函数，而 u 是 x 的函数时：dy/dx = (dy/du) × (du/dx)。求 y = (3x^2 + 2)^5 的导数。令 u = 3x^2 + 2，则 y = u^5。那么 dy/du = 5u^4，du/dx = 6x。所以 dy/dx = 5(3x^2+2)^4 × 6x = 30x(3x^2+2)^4。
30| 30| 31| 31|Implicit differentiation extends the chain rule to equations where y cannot be easily isolated, such as x^2 + y^2 = 25 (a circle). Differentiate both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y. This technique is essential for connected rates of change problems in Pure 4.
32| 32| 33| 33|隐函数微分将链式法则扩展到 y 无法轻易分离的方程，例如 x^2 + y^2 = 25（圆）。对 x 两边求导，将 y 视为 x 的函数：2x + 2y(dy/dx) = 0，所以 dy/dx = -x/y。这个技巧对 Pure 4 中的相关变化率问题至关重要。
34| 34| 35| 35|Integration: The Reverse Journey 积分逆运算
36| 36| 37| 37|If differentiation gives the gradient, integration gives the area under the curve. The reverse power rule states: the integral of x^n dx = x^(n+1)/(n+1) + C (for n not equal to -1). The constant C is critical: forgetting it loses a mark in every A-Level integration question. For definite integrals with limits a and b, evaluate F(b) - F(a) where F is the antiderivative.
38| 38| 39| 39|如果微分给出梯度，积分则给出曲线下的面积。反向幂法则：x^n dx 的积分 = x^(n+1)/(n+1) + C（n 不等于 -1）。常数 C 至关重要：忘记它会在每道 A-Level 积分题中扣分。对于带上下限 a 和 b 的定积分，计算 F(b) - F(a)，其中 F 是反导数。
40| 40| 41| 41|Integration by substitution is the mirror of the chain rule. When you see a composite function, choose u as the inner function. For example, integrate 2x(x^2+1)^3 dx. Let u = x^2+1, so du/dx = 2x, meaning du = 2x dx. The integral becomes the integral of u^3 du = u^4/4 + C = (x^2+1)^4/4 + C. Always check your answer by differentiating!
42| 42| 43| 43|换元积分法是链式法则的镜像。当你看到复合函数时，选择 u 为内部函数。例如，积分 2x(x^2+1)^3 dx。令 u = x^2+1，则 du/dx = 2x，即 du = 2x dx。积分变为 u^3 du 的积分 = u^4/4 + C = (x^2+1)^4/4 + C。始终通过求导来检验答案！
44| 44| 45| 45|Integration by parts comes from the product rule reversed: the integral of u dv = uv - the integral of v du. Use the LIATE rule to choose u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. For the integral of x e^x dx, let u = x (algebraic) and dv = e^x dx. Then du = dx, v = e^x. Result: x e^x - the integral of e^x dx = x e^x - e^x + C = e^x(x-1) + C.
46| 46| 47| 47|分部积分法来自乘法法则的反向：u dv 的积分 = uv - v du 的积分。使用 LIATE 法则选择 u：对数、反三角、代数、三角、指数。对于 x e^x dx 的积分，令 u = x（代数优先），dv = e^x dx。则 du = dx，v = e^x。结果：x e^x - e^x dx 的积分 = x e^x - e^x + C = e^x(x-1) + C。
48| 48| 49| 49|Partial fractions decompose rational functions before integration. For (3x+5)/((x+1)(x+2)), write it as A/(x+1) + B/(x+2). Multiply through: 3x+5 = A(x+2) + B(x+1). Substitute x = -2: -6+5 = B(-1), so B = 1. Substitute x = -1: -3+5 = A(1), so A = 2. The integral becomes the integral of [2/(x+1) + 1/(x+2)] dx = 2 ln|x+1| + ln|x+2| + C.
50| 50| 51| 51|部分分式在积分前分解有理函数。对于 (3x+5)/((x+1)(x+2))，写为 A/(x+1) + B/(x+2)。通分：3x+5 = A(x+2) + B(x+1)。代入 x = -2：-6+5 = B(-1)，B = 1。代入 x = -1：-3+5 = A(1)，A = 2。积分变为 [2/(x+1) + 1/(x+2)] dx 的积分 = 2 ln|x+1| + ln|x+2| + C。
52| 52| 53| 53|Applications: Tangents, Normals, and Optimisation 应用：切线、法线和优化
54| 54| 55| 55|The derivative at a point gives the gradient of the tangent line. The equation of the tangent at x = a is y - f(a) = f'(a)(x - a). The normal is perpendicular, so its gradient is -1/f'(a). Edexcel Paper 1 often includes a 6-mark question combining differentiation, tangent/normal equations, and intersection points.
56| 56| 57| 57|函数在某点的导数给出切线的梯度。在 x = a 处的切线方程是 y - f(a) = f'(a)(x - a)。法线垂直于切线，其梯度为 -1/f'(a)。Edexcel Paper 1 经常包含一道 6 分题，结合微分、切线/法线方程和交点。
58| 58| 59| 59|Turning points occur where dy/dx = 0. Classify them using the second derivative: if d^2y/dx^2 > 0, it is a minimum; if d^2y/dx^2 < 0, it is a maximum; if d^2y/dx^2 = 0, check the sign change of dy/dx on either side. For optimisation problems (maximising area, minimising cost), set up an equation, differentiate, set dy/dx = 0, solve, and use the second derivative test to confirm.
60| 60| 61| 61|拐点出现在 dy/dx = 0 处。使用二阶导数分类：如果 d^2y/dx^2 > 0，则为极小值；d^2y/dx^2 < 0 则为极大值；若 d^2y/dx^2 = 0，则检查 dy/dx 在两侧的符号变化。对于优化问题（最大面积、最小成本），建立方程，求导，令 dy/dx = 0，求解，并用二阶导数检验确认。
62| 62| 63| 63|Definite Integration and Area 定积分与面积
64| 64| 65| 65|The definite integral from a to b of f(x) dx gives the signed area between the curve and the x-axis. Areas below the axis are negative! Edexcel loves trick questions where you must split the integral at x-intercepts. For y = x^3 - 4x between x = -2 and x = 2, the function crosses the axis at x = -2, 0, 2. You must compute the area as: |integral from -2 to 0| + |integral from 0 to 2|.
66| 66| 67| 67|
从 a 到 b 的定积分 f(x) dx 给出曲线与 x 轴之间的带符号面积。轴下方的面积为负！Edexcel 喜欢出陷阱题，你必须在 x 轴交点处分割积分。对于 y = x^3 - 4x，在 x = -2 到 x = 2 之间，函数在 x = -2, 0, 2 处穿过 x 轴。必须计算面积：|从 -2 到 0 的积分| + |从 0 到 2 的积分|。
68| 68| 69| 69|Area between two curves: the integral from a to b of [f(x) - g(x)] dx, where f(x) is the upper curve. Always sketch the graph first. Find intersection points by solving f(x) = g(x). For the area between y = x^2 and y = x+2, the curves intersect at x = -1 and x = 2. The upper curve is y = x+2, so area = integral from -1 to 2 of (x+2 - x^2) dx = [x^2/2 + 2x - x^3/3] from -1 to 2 = 9/2.
70| 70| 71| 71|两曲线之间的面积：从 a 到 b 的 [f(x) - g(x)] dx 的积分，其中 f(x) 是上方曲线。务必先画草图。通过解 f(x) = g(x) 找到交点。对于 y = x^2 和 y = x+2 之间的面积，曲线在 x = -1 和 x = 2 处相交。上方曲线是 y = x+2，面积 = 从 -1 到 2 的 (x+2 - x^2) dx 的积分 = [x^2/2 + 2x - x^3/3] 从 -1 到 2 = 9/2。
72| 72| 73| 73|Parametric Differentiation 参数微分
74| 74| 75| 75|When x and y are both functions of t (a parameter), use dy/dx = (dy/dt) / (dx/dt). For the parametric equations x = t^2 + 1, y = t^3 - t: dx/dt = 2t, dy/dt = 3t^2 - 1, so dy/dx = (3t^2 - 1)/(2t). To find the equation of the tangent at a specific point, first find t, then dy/dx at that t-value. Parametric integration is equally important: the area under a parametric curve is the integral of y (dx/dt) dt.
76| 76| 77| 77|当 x 和 y 都是 t 的函数（参数）时，使用 dy/dx = (dy/dt) / (dx/dt)。对于参数方程 x = t^2 + 1, y = t^3 - t：dx/dt = 2t, dy/dt = 3t^2 - 1，所以 dy/dx = (3t^2 - 1)/(2t)。要在特定点求切线方程，先求 t，再求该 t 值处的 dy/dx。参数积分同样重要：参数曲线下的面积是 y (dx/dt) dt 的积分。
78| 78| 79| 79|Trapezium Rule and Numerical Methods 梯形法则与数值方法
80| 80| 81| 81|When an integral cannot be evaluated analytically, Edexcel expects you to use the trapezium rule: area is approximately (h/2)[y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1})], where h = (b-a)/n is the strip width. The question will specify n, the number of strips. Always state whether your estimate is an overestimate or underestimate by checking the curve's concavity: if the curve is convex (d^2y/dx^2 > 0), the trapezium rule overestimates.
82| 82| 83| 83|当积分无法解析求解时，Edexcel 希望你使用梯形法则：面积约等于 (h/2)[y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1})]，其中 h = (b-a)/n 是条带宽度。题目会指定 n 的值。通过检查曲线的凹凸性来判断你的估计是偏高还是偏低：如果曲线是凸的 (d^2y/dx^2 > 0)，梯形法则会高估。
84| 84| 85| 85|Exponential and Natural Logarithm 指数与自然对数
86| 86| 87| 87|The derivative of e^x is e^x itself, and the integral of e^x dx is e^x + C. This unique property makes e^x the most important function in calculus. For e^(kx), dy/dx = k e^(kx), and the integral of e^(kx) dx = (1/k) e^(kx) + C. The natural logarithm is the inverse: d/dx [ln x] = 1/x (for x > 0), and the integral of 1/x dx = ln|x| + C.
88| 88| 89| 89|e^x 的导数就是 e^x 自身，e^x dx 的积分是 e^x + C。这个独特性质让 e^x 成为微积分中最重要的函数。对于 e^(kx)：dy/dx = k e^(kx)，e^(kx) dx 的积分 = (1/k) e^(kx) + C。自然对数是其逆运算：d/dx [ln x] = 1/x（x > 0），1/x dx 的积分 = ln|x| + C。
90| 90| 91| 91|Differential equations using e^x appear frequently: if dy/dx = ky, then y = A e^(kx). This models exponential growth (k > 0, like population) and decay (k < 0, like radioactive decay). Edexcel often pairs this with a context: Newton's Law of Cooling, compound interest, or chemical reaction rates. Always use the given initial condition to find the constant A.
92| 92| 93| 93|涉及 e^x 的微分方程频繁出现：如果 dy/dx = ky，则 y = A e^(kx)。这模拟指数增长（k > 0，如人口）和衰减（k < 0，如放射性衰变）。Edexcel 经常将此与具体情境配对：牛顿冷却定律、复利或化学反应速率。始终使用给定初始条件求常数 A。
94| 94| 95| 95|Trigonometric Calculus 三角微积分
96| 96| 97| 97|Memorise these six results. Derivatives: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[tan x] = sec^2 x. Integrals: the integral of cos x dx = sin x + C, the integral of sin x dx = -cos x + C, the integral of sec^2 x dx = tan x + C. For sin(kx) and cos(kx), apply the reverse chain rule: divide by k. The integral of sin(3x) dx = -(1/3) cos(3x) + C.
98| 98| 99| 99|熟记这六个结果。导数：d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[tan x] = sec^2 x。积分：cos x dx 的积分 = sin x + C，sin x dx 的积分 = -cos x + C，sec^2 x dx 的积分 = tan x + C。对于 sin(kx) 和 cos(kx)，应用反向链式法则：除以 k。sin(3x) dx 的积分 = -(1/3) cos(3x) + C。
100| 100| 101| 101|Trigonometric identities are indispensable for integration. To integrate sin^2 x, use the double-angle identity sin^2 x = (1 - cos 2x)/2. Then the integral of sin^2 x dx = the integral of (1/2 - cos 2x / 2) dx = x/2 - sin 2x / 4 + C. Similarly, cos^2 x = (1 + cos 2x)/2. These substitutions appear in nearly every Edexcel Pure 3 paper.
102| 102| 103| 103|三角恒等式对积分不可或缺。要积分 sin^2 x，使用倍角恒等式 sin^2 x = (1 - cos 2x)/2。那么 sin^2 x dx 的积分 = (1/2 - cos 2x / 2) dx 的积分 = x/2 - sin 2x / 4 + C。类似地，cos^2 x = (1 + cos 2x)/2。这些代换几乎出现在每份 Edexcel Pure 3 试卷中。
104| 104| 105| 105|Exam Technique: Common Pitfalls 考试技巧：常见失分点
106| 106| 107| 107|Pitfall 1: Forgetting +C in indefinite integration. This costs one mark every time. Make it a habit to write +C immediately after your antiderivative.
108| 108| 109| 109|失分点 1：不定积分忘记写 +C。每次都会扣一分。养成反导数后面立即写 +C 的习惯。
110| 110| 111| 111|Pitfall 2: Sign errors when differentiating cos x and sin x. The derivative of cos x is negative sin x. A mnemonic: "sin goes to cos, cos goes to minus sin."
112| 112| 113| 113|失分点 2：求 cos x 和 sin x 导数时的符号错误。cos x 的导数是负的 sin x。记忆口诀：sin 变 cos，cos 变负 sin。
114| 114| 115| 115|Pitfall 3: Forgetting to divide by the derivative of the inner function when integrating composite functions. The integral of (2x+1)^5 dx is NOT (2x+1)^6/6 + C. It is (2x+1)^6/12 + C because you must divide by 2. Always differentiate your answer to check.
116| 116| 117| 117|失分点 3：积分复合函数时忘记除以内部函数的导数。(2x+1)^5 dx 的积分不是 (2x+1)^6/6 + C，而是 (2x+1)^6/12 + C，因为你必须除以 2。始终通过求导检验答案。
118| 118| 119| 119|Pitfall 4: In area problems, not checking whether the curve crosses the x-axis. When it does, the definite integral gives "net area" (positive minus negative), not total area. Always sketch the curve and split at roots.
120| 120| 121| 121|失分点 4：面积问题中未检查曲线是否穿过 x 轴。如果穿过，定积分给出的是"净面积"（正值减负值），而非总面积。务必画曲线草图并在根处分割。
122| 122| 123| 123|Pitfall 5: Mixing up the product and quotient rules. The numerator of the quotient rule is v du/dx - u dv/dx, NOT u dv/dx - v du/dx. If you mix them up, your sign flips and the whole answer is wrong.
124| 124| 125| 125|失分点 5：混淆乘法法则和除法法则。除法法则的分子是 v du/dx - u dv/dx，不是 u dv/dx - v du/dx。混淆了符号就会翻转，整个答案都错了。
126| 126| 127| 127| 128| 128|🎓 Need Help with A-Level Maths? 需要 A-Level 数学辅导？
129| 129|Struggling with calculus? Our experienced tutors break down every topic step by step, from basic differentiation to advanced integration. Whether it is chain rule confusion or integration by parts paralysis, we have helped hundreds of Edexcel students achieve A* grades.
130| 130|微积分有困难？我们的资深导师逐步拆解每个知识点，从基础微分到高级积分。无论是链式法则的困惑还是分部积分的卡壳，我们已经帮助数百名 Edexcel 学生取得 A* 成绩。
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