📚 Differentiation and Integration: Core Techniques and Applications | 微分与积分：核心技巧与应用
Differentiation and integration are the two central operations of calculus. A-level mathematics papers use them in a wide range of contexts, from finding gradients and turning points to calculating areas and solving optimisation problems. This article reviews the essential rules, common applications and exam techniques covered in many pure mathematics courses.
微分和积分是微积分的两个核心运算。A-level 数学试卷在各类情境中都会涉及它们，从求斜率和转折点到计算面积以及解决最优化问题。本文回顾纯数学课程中常见的基本法则、常见应用和考试技巧。
1. The Derivative and Its Meaning | 导数及其意义
f(x) with respect to x. It tells you how quickly y changes when x changes, and it is the central idea behind gradients, velocities and marginal rates.
导数衡量函数 y = f(x) 相对于 x 的瞬时变化率。它表示当 x 变化时 y 变化得有多快，是斜率、速度与边际变化率背后的核心概念。
It is written as f'(x), dy/dx or d/dx[f(x)]. These notations are interchangeable in most A-level questions, although dy/dx is especially useful when y is defined implicitly or parametrically.
它写作 f'(x)、dy/dx 或 d/dx[f(x)]。在大多数 A-level 题目中这些记号可以互换，不过当 y 由隐函数或参数方程定义时，dy/dx 尤其有用。
Geometrically, f' ivative means the curve is rising at that point, while a negative derivative means it is falling.
从几何上看，f'(a) 是曲线 y = f(x) 在 x = a 处切线的斜率。导数为正表示曲线在该点上升，导数为负表示曲线在该点下降。
The derivative is defined by the limit below, provided this limit exists. If the limit does not exist at a point, the function is not differentiable there, which may happen at corners, cusps or discontinuities.
导数的定义如下面的极限，前提是该极限存在。如果某点极限不存在，函数在该点不可导，这可能发生在尖角、尖点或不连续处。
f'(x) = lim (h→0) [f(x + h) − f(x)] / h
Common notations include dy/dx, f'(x), y' and Leibniz notation d/dx. In A-level problems, you are often given y in terms of x and asked to find dy/dx, or given a point and asked for the gradient there.
常见记号包括 dy/dx、f'(x)、y' 以及莱布尼茨记号 d/dx。在 A-level 题目中，通常会给出 y 关于 x 的表达式，要求求出 dy/dx，或者给出一点并要求该点处的斜率。
2. Basic Differentiation Rules | 基本微分法则
xⁿ⁻¹. This works for positive integers, negative powers and fractional powers, making it the most frequently used rule in A-level calculus.
对于任意常数 c 和实数 n，幂法则指出 d/dx[xⁿ] = n xⁿ⁻¹。这适用于正整数、负指数和分数指数，因此是 A-level 微积分中最常用的法则。
ve of a constant is zero: d/dx[c] = 0. The constant multiple rule gives d/dx[c f(x)] = c f' low term-by-term differentiation, so polynomials can be differentiated one term at a time.
常数的导数为零：d/dx[c] = 0。常数倍法则给出 d/dx[c f(x)] = c f'(x)。和差法则允许逐项求导，因此多项式可以逐项微分。
Common derivatives to memorise include d/dx[sin x] = cos x, d/dx[cos x] = −sin x, d/dx[eˣ] = eˣ n x] = 1/x for x > 0. These appear repeatedly in pure mathematics and mechanics questions.
需要熟记的常见导数包括 d/dx[sin x] = cos x，d/dx[cos x] = −sin x，d/dx[eˣ] = eˣ，以及 d/dx[ln x] = 1/x（x > 0）。这些在纯数学和力学题目中反复出现。
Before differentiating, rewrite expressions using index laws. For example, 1/x² becomes x⁻², and √x becomes x1/2. This converts fractions and roots into powers, so the power rule can be applied directly.
求导前先用指数法则改写表达式。例如，1/x² 写成 x⁻²，√x 写成 x1/2。这样可以把分式和根式转化为幂，从而直接使用幂法则。
| Function | 函数 | Derivative | 导数 |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| eˣ | eˣ |
| ln x | 1/x, x > 0 |
3. Product, Quotient and Chain Rules | 乘积、商与链式法则
n two functions are multiplied: d/dx[uv] = u dv/dx + v du/dx. It is often written as (u v)' = u'v + uv'. A common mistake is to differentiate each factor separately, which is incorrect.
当两个函数相乘时使用乘积法则：d/dx[uv] = u dv/dx + v du/dx。它也常写作 (u v)' = u'v + uv'。常见错误是分别对每个因式求导，这是不正确的。
/ v². The numerator has a minus sign, and the order matters. Many students remember it as low d high minus high d low over low squared.
商法则适用于分式 u/v：d/dx[u/v] = (v du/dx − u dv/dx) / v²。分子中是减号，且顺序很重要。可以记作分母乘分子导数减去分子乘分母导数，再除以分母的平方。
or example, if y = (3x + 1)⁵, let u = 3x + 1, so y = u⁵ and dy/dx = 5u⁴ × 3 = 15(3x + 1)⁴.
链式法则处理复合函数：dy/dx = dy/du × du/dx。例如，若 y = (3x + 1)⁵，令 u = 3x + 1，则 y = u⁵，所以 dy/dx = 5u⁴ × 3 = 15(3x + 1)⁴。
ears with trigonometric, exponential and logarithmic functions, such as y = sin(2x), y = e3x or y = ln(5x + 2). Always identify the inner function before differentiating.
考试中链式法则常与三角、指数和对数函数一起出现，例如 y = sin(2x)、y = e3x 或 y = ln(5x + 2)。求导前务必先识别内层函数。
4. Higher Derivatives and Rates of Change | 高阶导数与变化率
The second derivative is the derivative of the first derivative. It is written as f''(x), y'' or d²y/dx². It measures how the gradient itself is changing, which is useful for identifying concavity and classifying stationary points.
二阶导数是导数的导数。它写作 f''(x)、y'' 或 d²y/dx²。它衡量斜率本身如何变化，可用于判断凹凸性和分类驻点。
If f''(x) > 0, the curve is concave upwards and the gradient is increasing. If f''(x) < 0, the curve is concave downwards and the gradient is decreasing. A change in concavity may indicate a point of inflection.
如果 f''(x) > 0，曲线向上凹，斜率递增。如果 f''(x) < 0，曲线向下凹，斜率递减。凹凸性改变可能表示拐点。
In kinematics, if s(t) is displacement, then ds/dt is velocity and d²s/dt² is acceleration. This connects calculus directly to motion questions involving variable speed, maximum velocity and distance travelled.
在运动学中，若 s(t) 是位移，则 ds/dt 是速度，d²s/dt² 是加速度。这把微积分直接联系到涉及变速度、最大速度和路程的运动问题。
s often involve expanding circles, leaking tanks or changing volumes with respect to time.
相关变化率问题使用链式法则形式 dA/dt = dA/dr × dr/dt。这类问题常涉及膨胀的圆、漏水的容器或随时间变化的体积。
5. Tangents, Normals and Stationary Points | 切线、法线与驻点
The tangent to a curve at a point has gradient m = f' tandard application of differentiation and appears in almost every pure mathematics paper.
曲线在某点的切线斜率为 m = f'(x₁)。其方程可写作 y − y₁ = m(x − x₁)。这是导数的标准应用，几乎每份纯数学试卷都会出现。
d m ≠ 0. If the tangent is horizontal, the normal is vertical, and its equation is x = x₁.
法线与切线垂直，因此其斜率为 −1/m，前提是 m ≠ 0。如果切线水平，则法线竖直，其方程为 x = x₁。
A stationary point occurs where f'(x) = 0. To classify it, use the second derivative test: f''(x) > 0 gives a local minimum, f''(x) < 0 gives a local maximum, and f''(x) = 0 requires further investigation.
驻点出现在 f'(x) = 0 处。分类时使用二阶导数判别法：f''(x) > 0 为局部极小值，f''(x) < 0 为局部极大值，f''(x) = 0 则需要进一步判断。
For f''(x) = 0, check the sign of f' to positive, it is a minimum; if it does not change sign, it may be a point of inflection.
当 f''(x) = 0 时，检查该点两侧 f'(x) 的符号。如果斜率由正变负，则为极大值；由负变正，则为极小值；如果符号不变，则可能是拐点。
6. Optimisation and Practical Modelling | 最优化与实际建模
Optimisation problems ask you to maximise or minimise a quantity such as area, volume, cost or profit. The general method is to form an equation for the quantity, differentiate it, set the derivative to zero and solve.
最优化问题要求最大化或最小化面积、体积、成本或利润等量。通用方法是先建立该量的方程，求导，令导数为零并求解。
If two variables appear, use the constraint to eliminate one of them. For example, if the perimeter of a rectangle is fixed, express the area in terms of one side only before differentiating.
如果出现两个变量，利用约束条件消去其中一个。例如，若矩形周长固定，先用一条边表示面积，然后再求导。
After finding the stationary point, justify that it is a maximum or minimum. You can use the second derivative, a sign table, or the nature of the physical situation. Always answer the question in context, including units.
找到驻点后，要证明它是最大值还是最小值。可以使用二阶导数、符号表或实际情境的性质。最后要结合情境作答，并带上单位。
Common examples include maximising the volume of an open box, minimising the surface area of a cylinder, and finding the minimum cost of fencing a field. These require careful algebraic setup before calculus is used.
常见例子包括最大化开口盒子的体积、最小化圆柱的表面积，以及求围栏一块田地的最低成本。这些都需要在运用微积分前仔细建立代数关系。
7. Integration as Reverse Differentiation | 积分作为微分的逆运算
Integration is the reverse process of differentiation. If F' re c is the constant of integration. This is called an indefinite integral because it represents a family of curves.
积分是微分的逆过程。如果 F'(x) = f(x)，则 ∫ f(x) dx = F(x) + c，其中 c 是积分常数。这称为不定积分，因为它表示一族曲线。
You can always check an integral by differentiating your answer. If the derivative returns the original integrand, the integration is correct up to the constant c. This is one of the most reliable exam checks.
你总可以通过求导来检验积分。如果导数回到原来的被积函数，那么积分在相差常数 c 的意义下是正确的。这是考试中最可靠的检查方法之一。
The constant of integration is essential for indefinite integrals. Omitting it can lose marks. In applied problems, an initial condition is usually given so that c can be found and a particular solution obtained.
对于不定积分，积分常数必不可少。漏写会失分。在应用问题中，通常会给出初始条件，以便求出 c 并得到特解。
8. Standard Integrals and the Constant of Integration | 标准积分与积分常数
ation power rule, and it works for positive, negative and fractional powers except n = −1.
积分幂法则为 ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c，其中 n ≠ −1。这是微分幂法则的逆运算，适用于正、负和分数指数，但 n = −1 除外。
When n = −1, the integral is ∫ 1/x dx = ln|x| use ln x is only defined for x > 0, while 1/x is defined for both positive and negative x.
当 n = −1 时，积分为 ∫ 1/x dx = ln|x| + c。绝对值很重要，因为 ln x 只在 x > 0 时有定义，而 1/x 在正负 x 上都有定义。
Standard integrals include ∫ eˣ dx = eˣ c, and ∫ sec² x dx = tan x + c. These should be recognised immediately in exam questions.
标准积分包括 ∫ eˣ dx = eˣ + c，∫ sin x dx = −cos x + c，∫ cos x dx = sin x + c，以及 ∫ sec² x dx = tan x + c。考试中应能立即识别这些积分。
More complex integrals are often reduced to standard forms by algebraic manipulation, such as expanding brackets, dividing terms, or using trigonometric identities before integrating.
更复杂的积分通常通过代数变形化为标准形式，例如展开括号、逐项除法，或在积分前使用三角恒等式。
9. Definite Integrals, Areas and the Fundamental Theorem | 定积分、面积与微积分基本定理
b) − F(a), where F is an antiderivative of f. This result is the Fundamental Theorem of Calculus and links differentiation with integration.
定积分有上下限，并给出一个数值：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的一个原函数。这一结果就是微积分基本定理，它把微分与积分联系起来。
The definite integral can represent the signed area between the curve and the x-axis. Areas above the x-axis are positive, while areas below the x-axis are negative. For total area, split the integral at the x-intercepts and take absolute values.
定积分可以表示曲线与 x 轴之间的有向面积。x 轴上方的面积为正，下方的面积为负。若要计算总面积，应在 x 轴截点处分段积分并取绝对值。
(x) on the interval. Always identify which curve is on top before setting up the integral.
两条曲线 y = f(x) 与 y = g(x) 在 x = a 到 x = b 之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx，前提是在该区间上 f(x) ≥ g(x)。列式前先判断哪条曲线在上方。
₁) + yₙ], where h = (b − a)/n. It is useful when the function cannot be integrated easily.
梯形法用直线段近似定积分：∫ₐᵇ y dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]，其中 h = (b − a)/n。当函数不易积分时，这个方法很有用。
10. Integration Techniques: Substitution, Parts and Partial Fractions | 积分技巧：换元、分部与部分分式
Integration by substitution reverses the chain rule. If u = g(x), then du = g'(x) dx, and ∫ f(g(x)) g' integrating, substitute back to the original variable unless the limits have been changed.
换元积分法是链式法则的逆用。若 u = g(x)，则 du = g'(x) dx，且 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。积分后要换回原变量，除非已经更改了上下限。
For definite integrals, change the limits to u-values when you change the variable. This avoids substituting back and reduces the chance of arithmetic errors at the end of a long question.
对于定积分，换元时要把上下限也改成 u 的值。这样可以避免最后再换回原变量，也能减少长题末尾的算术错误。
v du. Choose u to be the part that simplifies when differentiated, such as ln x, x or x².
分部积分法是乘积法则的逆用：∫ u dv/dx dx = uv − ∫ v du/dx dx。它也常写作 ∫ u dv = uv − ∫ v du。选择 u 时，应选求导后变简单的部分，例如 ln x、x 或 x²。
Partial fractions are used for rational integrals. Split a fraction such as 1/[(x + 1)(x + 2)] into A/(x + 1) + B/(x + 2), then integrate each term separately, often producing logarithmic terms.
部分分式用于有理函数积分。把 1/[(x + 1)(x + 2)] 这样的分式拆成 A/(x + 1) + B/(x + 2)，然后逐项积分，结果通常包含对数项。
11. Differential Equations and Applied Contexts | 微分方程与应用情境
A differential
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