Calculus Core Formulas Revision | 微积分核心公式梳理

📚 Calculus Core Formulas Revision | 微积分核心公式梳理

Calculus is one of the most heavily weighted topics in the A-Level Mathematics syllabus. Mastering the core formulas — from differentiation rules to integration techniques — is essential for solving exam problems efficiently and accurately. This article consolidates the key formulas you must know, organised by topic for quick revision.

微积分是A-Level数学考试中占比最重的内容之一。熟练掌握核心公式——从微分法则到积分技巧——是高效、准确解题的关键。本文将按主题整理你必备的核心公式，便于快速复习。

1. Differentiation Basics | 微分基础法则

The power rule states that for a function \(y = x^n\), the derivative is given by multiplying by the power and reducing it by one.

幂函数法则指出，对于函数 \(y = x^n\)，其导数为乘以指数并将指数减一。

If y = xⁿ, then dy/dx = n·xⁿ⁻¹

For a constant multiplied by a function, the constant factor is preserved during differentiation. The derivative of a sum or difference of functions is simply the sum or difference of their individual derivatives.

常数乘以函数的导数保持不变；函数和或差的导数等于各函数导数的和或差。

d/dx [c·f(x)] = c·f'(x) | d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

You must also remember that the derivative of a constant is zero. For example, if y = 5, then dy/dx = 0. When differentiating expressions involving fractional or negative indices, rewrite them in index form first. For instance, √x = x^(1/2) and 1/x = x⁻¹.

另外务必记住常数的导数为零。例如，若 y = 5，则 dy/dx = 0。当处理含分数指数或负指数的表达式时，先将其改写为指数形式。例如，√x = x^(1/2)，1/x = x⁻¹。

2. Product and Quotient Rules | 乘积法则与商法则

The product rule is used when differentiating the product of two functions. It states that the derivative is the first function times the derivative of the second, plus the second function times the derivative of the first.

乘积法则用于两个函数相乘的求导。其表述为：导数等于第一个函数乘以第二个函数的导数，再加上第二个函数乘以第一个函数的导数。

If y = u·v, then dy/dx = u·(dv/dx) + v·(du/dx)

The quotient rule applies when one function is divided by another. It states that the derivative is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all over the denominator squared.

商法则适用于一个函数除以另一个函数的情形。其表述为：导数等于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。

If y = u/v, then dy/dx = [v·(du/dx) − u·(dv/dx)] / v²

A common mnemonic for the product rule is ‘left d right plus right d left’, and for the quotient rule, ‘low d high minus high d low, over low squared’. Always check whether you can simplify the function first — sometimes a term can be split into separate fractions, allowing you to avoid the quotient rule entirely.

乘积法则的口诀为“左导右 + 右导左”，商法则的口诀为“低导高 − 高导低，除以低平方”。解题前先检查能否化简原函数——有时可将项拆分为多个独立的分式，从而完全避免使用商法则。

3. Chain Rule | 链式法则

The chain rule is essential for differentiating composite functions — functions of functions. If a variable u depends on x, and y depends on u, then the derivative of y with respect to x is the product of the two separate derivatives.

链式法则是复合函数求导的关键工具。若变量 u 依赖于 x，且 y 依赖于 u，则 y 对 x 的导数等于两个独立导数的乘积。

If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx)

In practice, when differentiating a function like y = (3x² + 1)⁵, treat the inner expression as u. Differentiate the outer power first, then multiply by the derivative of the inner bracket.

实际应用中，当求导形如 y = (3x² + 1)⁵ 的函数时，将内部表达式视为 u。先对外层幂求导，再乘以内部括号的导数。

y = (ax + b)ⁿ → dy/dx = n·a·(ax + b)ⁿ⁻¹

The chain rule also applies to trigonometric functions and exponentials. For instance, if y = sin(kx), then dy/dx = k·cos(kx); if y = eᵏˣ, then dy/dx = k·eᵏˣ.

链式法则同样适用于三角函数和指数函数。例如，若 y = sin(kx)，则 dy/dx = k·cos(kx)；若 y = eᵏˣ，则 dy/dx = k·eᵏˣ。

4. Derivatives of Standard Functions | 常见函数的导数

The following table lists the derivatives of standard functions that appear frequently in A-Level exams. You should memorise these completely.

下表列出A-Level考试中高频出现的常见函数导数，请务必牢记。

Function \| 函数	Derivative \| 导数
xⁿ	n·xⁿ⁻¹
eˣ	eˣ
eᵏˣ	k·eᵏˣ
ln(x)	1/x
sin(x)	cos(x)
cos(x)	−sin(x)
tan(x)	sec²(x)
sin(kx)	k·cos(kx)
cos(kx)	−k·sin(kx)

Note that the derivative of ln(x) is only valid for x > 0. The derivatives of tan(x) and other trigonometric functions are derived from the quotient rule, but you should know the results directly.

注意 ln(x) 的导数仅在 x > 0 时成立。tan(x) 等三角函数的导数可由商法则推导，但建议直接记忆结果。

5. Integration Basics | 积分基础法则

Integration is the reverse process of differentiation. The power rule for integration states that to integrate xⁿ, you add one to the index and divide by the new index.

积分是微分的逆运算。幂函数的积分法则是：指数加一，再除以新的指数。

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

The constant C is the integration constant, which must always be included for indefinite integrals. For the special case n = −1, the integral of 1/x is the natural logarithm of the absolute value of x.

常数 C 为积分常数，不定积分中必须始终包含。特殊情形 n = −1 时，1/x 的积分为 x 绝对值的自然对数。

∫(1/x) dx = ln|x| + C

Integration is linear: the integral of a constant multiple is that constant times the integral, and the integral of a sum is the sum of the integrals. You may also need to rewrite expressions in index form before integrating, such as converting √x to x^(1/2).

积分具有线性性质：常数倍函数的积分等于常数乘以原函数的积分；函数和的积分等于各函数积分的和。有时需先将被积表达式改写为指数形式，例如将 √x 转换为 x^(1/2)。

6. Integrals of Standard Functions | 常见函数的积分

The table below lists indefinite integrals of standard functions. Memorise these results — they are the building blocks for all integration problems in the exam.

下表列出常见函数的不定积分，务必熟记——它们是解答考试中所有积分问题的基础。

Function \| 函数	Integral \| 积分
xⁿ	xⁿ⁺¹/(n+1) + C
1/x	ln\|x\| + C
eˣ	eˣ + C
eᵏˣ	(1/k)·eᵏˣ + C
cos(x)	sin(x) + C
sin(x)	−cos(x) + C
sec²(x)	tan(x) + C
cos(kx)	(1/k)·sin(kx) + C
sin(kx)	−(1/k)·cos(kx) + C

For integrals involving (ax + b)ⁿ, the standard form is obtained by dividing by the coefficient of x. For example, ∫(2x + 1)³ dx = (2x + 1)⁴/(4 × 2) + C = (2x + 1)⁴/8 + C.

对于涉及 (ax + b)ⁿ 的积分，标准形式为将结果除以 x 的系数。例如，∫(2x + 1)³ dx = (2x + 1)⁴/(4 × 2) + C = (2x + 1)⁴/8 + C。

7. Definite Integrals and Area | 定积分与面积

A definite integral is evaluated between two limits and produces a numerical value. To evaluate it, find the indefinite integral, substitute the upper limit, subtract the value at the lower limit.

定积分在两个界限之间求值，得出一个数值结果。计算时，先求出不定积分，代入上限值，减去下限值即可。

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)

The definite integral \(\int_a^b f(x)\,dx\) represents the signed area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b. If the function dips below the x-axis, the corresponding region contributes a negative area.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a 与 x = b 之间的有向面积。若函数图像位于 x 轴下方，则相应区域的面积贡献为负值。

Area = ∫ₐᵇ |f(x)| dx (if f(x) changes sign)

For the area between two curves y = f(x) and y = g(x), integrate the difference between the upper and lower functions. If f(x) ≥ g(x) on the interval [a, b], the area is given by ∫ₐᵇ [f(x) − g(x)] dx.

对于两条曲线 y = f(x) 与 y = g(x) 之间的面积，对上下函数的差进行积分。若在区间 [a, b] 上 f(x) ≥ g(x)，则面积为 ∫ₐᵇ [f(x) − g(x)] dx。

8. Area Under a Parametric Curve | 参数曲线下的面积

For curves defined parametrically by x = f(t) and y = g(t), the area under the curve cannot be integrated directly with respect to x. Instead, use the substitution dx = (dx/dt)·dt.

对于由 x = f(t)、y = g(t) 定义的参数曲线，其面积不能直接对 x 积分。应利用代换 dx = (dx/dt)·dt 来处理。

Area = ∫ y dx = ∫ y·(dx/dt) dt

When applying this formula, ensure the limits are the t-values corresponding to the required x-values, not the x-values themselves. The direction of integration matters: if t₂ > t₁ but y·(dx/dt) is negative, the integral may produce a negative value.

应用此公式时，确保积分限是对应的 t 值而非 x 值。积分方向很重要：若 t₂ > t₁ 但 y·(dx/dt) 为负，积分可能产生负值。

As an example, for the parametric curve x = t², y = t³, the area between t = 0 and t = 2 is ∫₀² t³·(2t) dt = ∫₀² 2t⁴ dt = [2t⁵/5]₀² = 64/5 square units.

例如，对于参数曲线 x = t²，y = t³，t 从 0 到 2 的面积为 ∫₀² t³·(2t) dt = ∫₀² 2t⁴ dt = [2t⁵/5]₀² = 64/5 平方单位。

9. Trapezium Rule | 梯形法则

When a function cannot be integrated analytically, or when data is only available at discrete points, the trapezium rule provides a numerical approximation for definite integrals.

当函数无法解析积分，或数据仅以离散点给出时，梯形法则为定积分提供数值近似方法。

∫ₐᵇ f(x) dx ≈ (h/2)·[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]

Here, h = (b − a)/n is the strip width, n is the number of equal intervals, and y₀, y₁, …, yₙ are the function values at equally spaced points. The formula treats each strip as a trapezium and sums their areas.

其中 h = (b − a)/n 为区间宽度，n 为等分区间数，y₀, y₁, …, yₙ 为等间距点上的函数值。该公式将每个小区间视为一个梯形，并对其面积求和。

In exam questions, you may be asked to estimate an integral using a given number of strips. When the curve is concave up, the trapezium rule overestimates the true area; when the curve is concave down, it underestimates. Increasing n improves accuracy but also increases computation.

考试中常要求使用给定条数估算积分值。曲线下凸时，梯形法则会高估真实面积；曲线下凹时会低估。增大 n 可提高精度，但也增加计算量。

10. Differential Equations and Applications | 微分方程与应用

A differential equation involves derivatives and describes how a quantity changes over time or space. The simplest type is of the form dy/dx = f(x), solved directly by integration.

微分方程涉及导数，描述某一量随时间或空间的变化规律。最简单的形式为 dy/dx = f(x)，直接积分即可求解。

For variables that can be separated, use the technique of separation of variables. Rearrange so all y-terms (including dy) are on one side and all x-terms (including dx) are on the other, then integrate both sides.

对于可分离变量的方程，使用变量分离法。将所有含 y 的项（含 dy）移到一边，含 x 的项（含 dx）移到另一边，然后对两边分别积分。

dy/dx = g(x)·h(y) → ∫(1/h(y)) dy = ∫g(x) dx

Applications of differential equations in A-Level include exponential growth and decay, Newton’s law of cooling, and simple population models. The general solution of dN/dt = kN is N = N₀eᵏᵗ, where N₀ is the initial value.

微分方程在A-Level中的应用包括指数增长与衰减、牛顿冷却定律以及简单种群模型。dN/dt = kN 的通解为 N = N₀eᵏᵗ，其中 N₀ 为初始值。

When solving differential equations with initial conditions, substitute the given values after integrating to find the particular constant C. Always include the constant of integration in the general solution stage.

求解带初值条件的微分方程时，在积分后代入已知条件求出常数 C。在通解阶段务必包含积分常数。

11. Gradients, Tangents and Normals | 切线、法线与梯度

The derivative dy/dx gives the gradient of the curve at any point. To find the equation of the tangent at a point (x₁, y₁), substitute x₁ into dy/dx to obtain the gradient m, then use the point-slope form of a line.

导数 dy/dx 给出曲线上任意点的斜率。求在点 (x₁, y₁) 处的切线方程时，将 x₁ 代入 dy/dx 得到斜率 m，然后使用点斜式直线方程即可。

Tangent: y − y₁ = m(x − x₁)

The normal is perpendicular to the tangent at the point of contact. The gradient of the normal is the negative reciprocal of the tangent’s gradient: m_normal = −1/m.

法线在接触点处与切线垂直。法线的斜率为切线斜率的负倒数：m_法线 = −1/m。

Normal: y − y₁ = (−1/m)(x − x₁)

Be careful when m = 0 — the tangent is horizontal, making the normal vertical (x = x₁). Similarly, if m is undefined, the tangent is vertical and the normal is horizontal.

注意当 m = 0 时，切线水平，法线垂直（x = x₁）。同理，若 m 不存在，则切线垂直，法线水平。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

When differentiating or integrating, always rewrite roots and reciprocal powers in index form first. This simple step prevents many algebraic errors.

求导或积分时，务必先将根式和倒数形式的幂改写为指数形式。这一步简单却能避免大量代数错误。

Do not forget the constant C for indefinite integrals. In definite integrals, the constant cancels out — but omitting C in an indefinite integral will cost marks.

不定积分中切勿遗漏常数 C。定积分中常数会相互抵消——但在不定积分中遗漏 C 会被扣分。

Check the sign of your answer when calculating areas. If a definite integral yields a negative value while you were asked for an area, take the absolute value or split the integral at the points where the curve crosses the x-axis.

计算面积时注意答案的符号。若定积分结果为负值而题目要求的是面积，则应取绝对值或在曲线与 x 轴交点处分段积分。

When applying the chain rule, always multiply by the derivative of the inner function. A common error is stopping after differentiating the outer function only.

使用链式法则时，务必乘以内层函数的导数。常见错误是在对外层函数求导后直接停止。

Finally, memorise the standard derivatives and integrals thoroughly. In the exam, these are not provided in the formula booklet for every board — the speed with which you recall them directly affects your time management.

最后，请彻底熟记标准导数和积分公式。在考试中，并非所有考试局都会在公式手册中提供这些——你回忆它们的速度直接影响时间管理。

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Calculus Core Formulas Revision | 微积分核心公式梳理