Edexcel数学 微积分 链式法则 换元 定积分
Edexcel A Level Mathematics（代码 9MA0，2017 年 9 月起施教，2019 年首次考试）把微积分编在 Topic 7 微分与 Topic 8 积分，两张纯数卷各 2 小时、100 分、各占 33.33%，所以微积分只出现在 Paper 1 与 Paper 2，卷面一共 200 分的空间。分值集中在三处：求导与积分规则本身、把导数用回切线与变化率、以及积分与面积和微分方程。很多学生以为微积分难在技巧，实际丢分多发生在读题后的第一步选择上，比如该用链式法则还是先展开再求导。下面按十个小节把首原理、链式法则、乘积与商法则、隐函数与参数方程、切线法线与驻点、相关变化率、定积分面积、换元与分部积分、微分方程、以及三周练习节奏逐条拆开。
Calculus occupies Topic 7 Differentiation and Topic 8 Integration in Pearson Edexcel A Level Mathematics, code 9MA0, first taught from September 2017 with first assessment in 2019, and both pure papers run for two hours at 100 marks each, which places calculus inside 200 of the 300 marks on offer. Marks are usually lost at the moment a method is chosen rather than in the algebra itself, so each section below works through the decision that comes before the working. The sequence runs from first principles to differential equations, with the mark scheme habits that decide whether a correct-looking answer scores.
首原理与导数定义
首原理求导在 9MA0 里是固定的小题，官方 Enhanced Content Guidance（Issue 2，2023 年 1 月）给出的例子是 x 的整数次幂。写法固定：先写 f(x+h)-f(x) 除以 h 的差商，展开时把同类项约掉，再用极限让 h 归零。学生最常把展开那一步做错，尤其是 (x+h)^3 这类式子，漏掉中间项里 3x 的系数就会一路错到答案。以 f(x)=x^2 为例，差商化简后是 2x+h，h 归零得到 2x，结论要写成 f'(x)=2x，而不是只留一个 2x 在卷面上。阅卷时这种小题通常 4 分左右，其中 1 分专门留给极限的表述，缺了 lim 或 h 趋于零那一行会扣掉它。最后一句还要落回几何意义，说明 x 处切线的斜率等于 2x，这一句常常是单独的 1 分。
Differentiation from first principles is set out in the Pearson Enhanced Content Guidance, issue 2 of January 2023, using integer powers of x as the worked example, and the method is always the same three moves: form the difference quotient, simplify before taking the limit, then let h fall to zero. The expansion is where candidates drop marks, especially on (x+h)^3, where a missing coefficient follows the answer all the way to the end. A full solution writes the limit notation explicitly and finishes with a sentence naming the gradient of the tangent, because that concluding line carries a mark of its own.
链式法则处理复合函数
复合函数的导数用链式法则，公式写成 dy/dx = dy/du 乘 du/dx，实际操作是把式子拆成外层与内层。拆分的判断很直接：能不能把整个表达式看成某个简单函数套在另一个函数里。以 y=(3x^2+1)^5 为例，外层是五次方，内层是 3x^2+1，导数就是 5(3x^2+1)^4 乘 6x，化简后是 30x(3x^2+1)^4。常见的错误是只写外层导数就收笔，把内层导数整个漏掉，这一步丢的是答案分而不是方法分。三角函数与指数函数混在一起时更容易出错，y=sin(2x) 的导数是 2cos(2x) 而不是 cos(2x)，y=e^(3x) 的导数是 3e^(3x)。阅卷时方法分通常先给，外层导数写对就有 1 分，所以先把链式结构写全再化简，比直接心算更容易保住分数。
The chain rule covers anything built from a function inside another function, written as dy/dx equal to dy/du times du/dx. Candidates most often differentiate the outer layer correctly and then stop, forgetting the inner derivative that the mark scheme rewards separately; writing the structure out in full before simplifying keeps both marks reachable. It matters even more with sin(2x) or e^(3x), where the inner factor changes the coefficient in the final answer.
乘积法则与商法则
乘积法则写成 u'v 加 uv'，两项都不能少，前后顺序不影响结果。识别信号是两项相乘且两项都不是常数，比如 y=x^2 e^x，取 u=x^2、v=e^x，得到 2x e^x 加 x^2 e^x，能提公因式就提出来写成 x e^x (2+x)。商法则写成 (u'v - uv') 除以 v 的平方，分子的减号是最容易写反的地方，写反后整题答案全错。学生最常把商法则的分子写成 u'v 加 uv'，因为记住了乘积法则的加号，这个错在纯数卷上很常见。判断用哪一个更省事有个实用标准：分母是单项式时先拆成负指数再求导，比硬套商法则快，例如 x^3 除以 x 这样；分母是二项式才值得套公式。阅卷时这两条法则通常 3 到 4 分，全部给在求导那一步。
The product rule keeps both terms, u'v plus uv', and the order does not matter; the quotient rule carries a minus sign in the numerator, and reversing it destroys the whole answer. A quick way to tell which rule is cheaper is to look at the denominator, because a single power can be turned into a negative index and differentiated directly, while a two-term denominator justifies the formula. These steps are worth three or four marks, all of them awarded at the differentiation line.
隐函数与参数方程求导
隐函数求导的核心是每一处 y 都要带上 dy/dx，因为 y 本身是 x 的函数。以 x^2 加 y^2 等于 25 为例，两边对 x 求导得到 2x 加 2y dy/dx 等于 0，整理后 dy/dx 等于 -x/y，把点的坐标代进去才得到具体斜率，例如点 (3,4) 处的斜率是 -3/4。参数方程用 dy/dx 等于 dy/dt 除以 dx/dt，分子分母分别对参数求导，最后代入 t 值。很多学生以为参数方程要先把 y 写成 x 的函数，其实那样多数题目做不下去，直接对 t 求导再相除就够了。常见的错误是隐函数里漏掉链式那一项，把 sin y 求导写成 cos y 而漏掉后面的 dy/dx。这一小节通常 5 到 6 分，切线方程与法线方程会接着用同一道题给出的点。
Implicit differentiation works because every y term carries a dy/dx factor, so x^2 plus y^2 equals 25 gives 2x plus 2y dy/dx equals zero and a gradient of minus x over y, which is -3/4 at the point (3,4). Parametric curves are handled with dy/dx equal to dy/dt divided by dx/dt, avoiding the impossible step of writing y as a function of x. The usual slip is a missing chain-rule factor, such as differentiating sin y as cos y without the dy/dx; the section is normally worth five or six marks and feeds the tangent or normal in the next part.
切线法线 驻点与二阶导
切线方程用 y - y1 = m(x - x1)，其中的 m 就是该点的导数值，法线的斜率取负倒数 -1/m。驻点满足一阶导等于零，判断性质用二阶导：二阶导大于零是极小值，小于零是极大值，得到的点要写成 (x, y) 坐标对。二阶导等于零的点不能直接下结论，要检查两侧一阶导的符号变化，例如 y=x^3 在原点既不是极大也不是极小。画一张表格是最稳的做法，把临界点与区间列出来，标上一阶导的正负号，答案不容易漏项，说明文字也有了依据。阅卷时这类题常见 8 到 10 分，其中求导 2 分、解方程 2 到 3 分、性质判断 2 分，剩下的给坐标与图像。学生最常把驻点坐标只写一个 x 值，题目要求给点就必须成对写出。
A tangent is built from y minus y1 equals m times x minus x1, with the gradient read from the derivative, while a normal takes the negative reciprocal. Stationary points come from a first derivative of zero, and the second derivative decides the type, positive for a minimum and negative for a maximum, so the answer has to be given as a coordinate pair. Where the second derivative is zero the sign of the first derivative on either side settles it, as y=x^3 does at the origin; a short table of intervals and signs preserves every mark.
相关变化率与小增量估计
相关变化率把两个变化量用链式法则连起来，题干通常给出体积或表面积随时间的变化，要求某个长度变化的快慢。做法是先把两个量之间的关系式写出来，再对时间 t 求导，例如球体积 V = 4/3 π r^3 给出 dV/dt 等于 4π r^2 dr/dt，已知 dV/dt 就能解出 dr/dt。单位要跟着写，厘米每秒、立方厘米每秒这类标注漏写会扣分。小增量估计是另一种考法，用 δy 约等于 f'(x) δx 估算变化量，题目一般要求给到三位有效数字。常见的错误是求导时忘记内层的变化率，把 dr/dt 当成 1，结果数值差出一个量级。这部分在纯数卷里通常 6 到 8 分，代入数值前先把符号写全，改起来更容易。
Connected rates of change join two quantities through the chain rule, typically giving a rate for volume or area and asking for the rate of a length. The relationship is written first and then differentiated with respect to time, so V = 4/3 pi r^3 becomes dV/dt = 4 pi r^2 dr/dt and the unknown follows by division; units belong in the answer, since an unlabelled rate loses an accuracy mark. Small increments use dy approximately f'(x) dx to three significant figures, and treating dr/dt as one is the error that shifts the answer by an order of magnitude.
定积分与阴影面积
定积分先求原函数再代上下限，写成 F(x) 从 a 到 b 等于 F(b) 减 F(a)，上下限代反会得到相反数。面积题要先看曲线是否穿过 x 轴，穿过时面积必须分段计算，直接积一个区间会把正负抵消掉。曲线与直线围成的区域，先联立求交点，再判断哪条在上方，用上方减下方积分，例如 y = 6x - x^2 与 y = 2x 的交点是 x 等于 0 和 x 等于 4，区间内曲线在上，积分得到 64/3。阅卷时积分过程单独给方法分，代限算错只扣答案分，所以即使数字不对也要把原函数写清楚。梯形法则用等宽小梯形近似面积，题目一般给表格数据，公式里第一个与最后一个纵坐标只取一半。这一组的整题常在 8 到 12 分，面积题还会要求给出单位。
A definite integral evaluates the primitive between the limits, so F(b) minus F(a) is the order that matters. Areas need care when the curve crosses the axis, because one integral covering both regions cancels the parts; for two curves the intersection points are found first and the upper expression is integrated minus the lower, which for y = 6x - x^2 against y = 2x gives 64/3 between zero and four. The trapezium rule approximates the same kind of area from a table of values, halving the first and last ordinates in the formula, and units usually belong in the final answer.
换元法与分部积分
换元法把积分化简成标准形式，关键是同时换掉被积表达式和 dx。以 ∫ 2x(x^2+1)^4 dx 为例，令 u = x^2+1，则 du = 2x dx，积分变成 ∫ u^4 du 等于 u^5/5 加 c，回代得到 (x^2+1)^5/5 加 c。定积分换元后上下限要跟着换成 u 的值，或者最后回代再代限，两种做法都成立，但必须做完其中一种，这是学生最常漏掉的一步。分部积分写成 ∫u dv 等于 uv 减 ∫v du，选择的经验是让求导后变简单的那一项当 u，ln x 与 x 的乘积通常先取 ln x。部分分式适用于分母能因式分解的有理函数，拆开后每一项都能直接积。这部分的小题分值常在 5 到 7 分之间，换元与回代各占一段。
Substitution rewrites an integral in a form that can be evaluated directly, and it has to change the differential as well as the expression; with u = x^2 + 1 the integral of 2x(x^2+1)^4 dx becomes the integral of u^4 du, giving (x^2+1)^5/5 plus c. Definite integrals need the limits converted or the substitution reversed, and skipping both is the commonest omission in this topic. Integration by parts follows the pattern uv minus the integral of v du, with the choice of u guided by which term simplifies when differentiated, while partial fractions handle rational functions whose denominator factorises.
微分方程与指数模型
微分方程通常可以分离变量，把含 y 的项移到一边、含 x 的项移到另一边再积分。以 dy/dx 等于 ky 为例，分离后得到 ln|y| 等于 kx 加 c，写成 y = Ae^(kx)，其中的 A 由初始条件确定。题目一般还会给出一个数据点，要求解出 k 的数值并解释它的含义，比如每年增长 4% 对应 k 约为 0.039，这一步要写清单位与时间基准。写成年增长却用月数据代入是常见的失分点，解释句因此不能含糊。阅卷时分离变量、积分、定常数、解释四段各有分，最后那句解释经常被整段跳过。这部分整题通常 8 到 12 分，是纯数卷里回报最高的一块。
Differential equations of this kind separate, so the y terms go one side and the x terms the other before integration, and dy/dx equal to ky leads to y = Ae^(kx) with A fixed by an initial condition. A data point is usually supplied so that the constant and its meaning can be stated, and an answer given in years while the data are monthly loses the interpretation mark even when the algebra is right. The closing explanatory sentence is the one candidates skip most often, and the whole question is normally worth eight to twelve marks.
三周练习与阅卷口径
三周练习可以按卷面结构排开。第一周每天两组小题，只练求导与积分的机械动作，链式法则、乘积与商法则、sin 与 cos、e^x 与 ln x 各写十题，每题限时 90 秒。第二周做整道大题，重点练读题后的第一步选择，把切线、阴影面积、相关变化率、微分方程四类题型各写五道的完整过程。第三周按 2 小时做一次完整的 Pure Mathematics 卷，用官方 mark scheme 自己批，重点看方法分有没有拿到。官方指南里的措辞可以直接当作信号，Show that 要求把中间步骤写全，Solutions relying entirely on calculator technology are not acceptable 意味着只写计算器结果拿不到分，Use algebraic integration 则要求用代数方法而不是图像读数。分数结构上，AO1 占 48% 到 52%，AO2 与 AO3 各占 23% 到 27%（Pearson 9MA0 spec），微积分的证明与建模小题正落在后两项里。
Three weeks of practice map onto the paper structure: mechanical drills for the chain, product and quotient rules with the standard derivatives, then whole questions chosen to rehearse the first decision rather than the algebra, then one full two-hour pure paper marked against the official scheme. The guidance phrases are signals worth memorising, since Show that demands every intermediate step in writing and Solutions relying entirely on calculator technology are not acceptable removes marks for answers lifted from a calculator. Weightings in the Pearson 9MA0 specification put AO1 at 48 to 52 per cent, with AO2 and AO3 each at 23 to 27 per cent, and proof and modelling questions in calculus sit in those latter bands.
海外直订Higher Mathematics for AQA GCSE 高等数学AQA GCSE¥645.0📱 去京东购买京东精选
需要一对一辅导？
A-Level / GCSE / IB / KS2-KS3 全科辅导，Mathematics 微积分可以安排求导与积分的专项计时练习
16621398022 同微信
查看 A-Level Mathematics 学习资源
或截图后用「扫一扫 → 相册」选择本图