ESAT Mathematics: Calculus & Differentiation | ESAT数学：微积分与微分专题

📚 ESAT Mathematics: Calculus & Differentiation | ESAT数学：微积分与微分专题

Calculus is the mathematics of change and motion, and differentiation sits at its very core. For ESAT candidates, mastering differentiation is not merely a syllabus requirement — it is the engine that powers physics kinematics questions, optimisation problems, and countless engineering applications. This guide walks through every differentiation technique you need, from first principles to implicit differentiation, with exam-specific insights throughout.

微积分是研究变化与运动的数学，而微分正是其核心。对于ESAT考生而言，掌握微分不仅是考纲要求，更是解决物理运动学问题、优化问题以及无数工程应用的关键引擎。本指南系统梳理你所需的一切求导技巧——从第一性原理到隐函数微分，全程贯穿ESAT考试要点。

1. What Is Differentiation? | 什么是微分？

Differentiation quantifies how a function changes as its input changes. Geometrically, the derivative f′(x) represents the slope of the tangent line to the curve y = f(x) at a given point x. Physically, it measures the instantaneous rate of change: velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity.

微分衡量的是函数随输入变化而变化的快慢。在几何上，导数 f′(x) 表示曲线 y = f(x) 在给定点 x 处切线的斜率。在物理上，它度量瞬时变化率：速度是位移对时间的导数，加速度是速度对时间的导数。

ESAT insight: Expect questions where a graph or a physical scenario requires you to identify which quantity is the derivative of another — velocity from a displacement-time graph is a classic example.

ESAT要点：考试中常出现这样的题目——根据图形或物理情境判断哪个量是另一个量的导数，位移-时间图像中的速度就是一个经典例子。

2. Limits and First Principles | 极限与第一性原理

The formal definition of the derivative is the limit of the average rate of change as the interval approaches zero. For a function f(x), the derivative at point x is defined as:

f′(x) = lim(h→0) [f(x+h) − f(x)] / h

For f(x) = x², we apply the definition directly:

f′(x) = lim(h→0) [(x+h)² − x²] / h = lim(h→0) (2xh + h²) / h = lim(h→0) (2x + h) = 2x

First-principles questions in ESAT typically involve expanding f(x+h), cancelling common factors, and then evaluating the limit. Practice this process for quadratic, cubic, and simple rational functions to build fluency.

ESAT中的第一性原理题通常需要展开 f(x+h)、消去公因式、再求极限。请针对二次、三次及简单有理函数反复练习这一流程，以建立熟练度。

Common mistake: forgetting that h is the variable approaching zero, not x. Keep x fixed while evaluating the limit in h.

常见错误：忘记趋近于零的变量是 h 而非 x。在求关于 h 的极限时，x 应被视为固定常数。

3. The Power Rule and Basic Operations | 幂法则与基本运算

If f(x) = xⁿ, then f′(x) = n·xⁿ⁻¹. This rule extends naturally to constant multiples and sums of functions: (cf)′ = c·f′ and (f + g)′ = f′ + g′. For example, if f(x) = 3x⁴ − 2x³ + x − 5, then f′(x) = 12x³ − 6x² + 1.

若 f(x) = xⁿ，则 f′(x) = n·xⁿ⁻¹。幂法则可自然推广到常数倍与函数之和：(cf)′ = c·f′，(f + g)′ = f′ + g′。例如，若 f(x) = 3x⁴ − 2x³ + x − 5，则 f′(x) = 12x³ − 6x² + 1。

The power rule also applies to negative and fractional exponents. For instance, f(x) = 1/x = x⁻¹ gives f′(x) = −x⁻² = −1/x², and f(x) = √x = x^(1/2) gives f′(x) = (1/2)x^(−1/2) = 1/(2√x).

幂法则同样适用于负指数和分数指数。例如，f(x) = 1/x = x⁻¹ 的导数为 f′(x) = −x⁻² = −1/x²；f(x) = √x = x^(1/2) 的导数为 f′(x) = (1/2)x^(−1/2) = 1/(2√x)。

ESAT tip: Rewrite functions in index form before differentiating — this turns seemingly complicated expressions into straightforward power-rule applications.

ESAT技巧：求导前先将函数改写为指数形式——这样看似复杂的表达式就变成了幂法则的直接应用。

4. Product Rule and Quotient Rule | 乘积法则与商法则

When two functions are multiplied, the derivative is not simply the product of their derivatives. The product rule states:

(fg)′ = f′g + fg′

For example, h(x) = x²·sin(x) gives h′(x) = 2x·sin(x) + x²·cos(x). Note that the order of differentiation — differentiating one factor at a time while keeping the other unchanged — is the essence of this rule.

当两个函数相乘时，导数并不是两者导数的简单乘积。乘积法则表明：

(fg)′ = f′g + fg′

例如，h(x) = x²·sin(x) 的导数为 h′(x) = 2x·sin(x) + x²·cos(x)。注意，该法则的本质是——每次只对一个因子求导，另一个因子保持不变，然后将它们相加。

For division, the quotient rule states that for f(x) = u/v:

(u/v)′ = (u′v − uv′) / v²

A useful mnemonic is “low d-high minus high d-low, over low squared”. For example, f(x) = x/(x² + 1):

对于除法，商法则表明，若 f(x) = u/v，则：

(u/v)′ = (u′v − uv′) / v²

一个实用的口诀是”上导下不导，减去下导上不导，除以分母平方”。例如，f(x) = x/(x² + 1)：

f′(x) = [1·(x² + 1) − x·2x] / (x² + 1)² = (1 − x²) / (x² + 1)²

ESAT practice: Product and quotient rules frequently appear alongside trigonometric and exponential functions — drill these combined questions to build speed.

ESAT练习：乘积法则和商法则经常与三角函数、指数函数结合出题——多练习这类综合题以提升速度。

5. The Chain Rule | 链式法则

The chain rule handles composite functions — functions inside functions. If y = f(u) and u = g(x), then:

dy/dx = dy/du × du/dx

Consider y = (3x² + 1)⁵. Let u = 3x² + 1. Then dy/du = 5u⁴ and du/dx = 6x, so:

链式法则用于处理复合函数——即函数嵌套函数的场景。若 y = f(u) 且 u = g(x)，则：

dy/dx = dy/du × du/dx

考虑 y = (3x² + 1)⁵。令 u = 3x² + 1，则 dy/du = 5u⁴，du/dx = 6x，因此：

dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴

The chain rule also works in reverse for differentiation of expressions like ln(sin x) or e^(x³). For y = e^(x³), let u = x³, giving dy/dx = 3x²·e^(x³).

链式法则同样适用于反向情形，如 ln(sin x) 或 e^(x³) 的求导。对于 y = e^(x³)，令 u = x³，得到 dy/dx = 3x²·e^(x³)。

ESAT tip: Recognise the “outer function–inner function” structure instantly. In multiple-choice questions, the correct answer often differs from distractors only by the chain-rule factor — a missing du/dx is the most common trap.

ESAT技巧：迅速识别”外层函数-内层函数”的结构。在选择题中，正确答案与干扰项的差别往往仅在于链式法则的因子——漏掉 du/dx 是最常见的陷阱。

6. Exponential, Logarithmic and Trigonometric Derivatives | 指数、对数与三角函数的导数

Beyond polynomials, ESAT requires fluency with the derivatives of standard functions. The table below summarises the essential results:

除多项式外，ESAT要求熟练掌握标准函数的导数。下表总结了核心求导公式：

f(x)	f′(x)
e^x	e^x
e^(kx)	k·e^(kx)
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec²x

Two key patterns emerge. First, the exponential function e^x is its own derivative — a unique and elegant property. Second, trigonometric derivatives involve a sign change: the derivative of cos x is −sin x, which often causes sign errors under time pressure.

这里呈现出两个关键规律。第一，指数函数 e^x 的导数是其自身——这是独一无二且优雅的性质。第二，三角函数的导数涉及符号变化：cos x 的导数是 −sin x，在时间压力下常导致符号错误。

For more complex combinations, these rules integrate with the chain rule: d/dx[sin(2x)] = 2cos(2x), and d/dx[ln(3x + 1)] = 3/(3x + 1).

对于更复杂的组合，这些公式需要与链式法则结合：d/dx[sin(2x)] = 2cos(2x)，d/dx[ln(3x + 1)] = 3/(3x + 1)。

ESAT practice: Many papers include a question testing d/dx[a·e^(bx)] or d/dx[ln(kx)] — these should be completed in under 30 seconds each.

ESAT练习：许多试卷中会出现 d/dx[a·e^(bx)] 或 d/dx[ln(kx)] 的题目——这类题应在30秒内完成。

7. Implicit and Parametric Differentiation | 隐函数与参数方程微分

Not all functions are given explicitly as y = f(x). Implicit differentiation handles equations like x² + y² = 25, where y is defined implicitly. Differentiate both sides with respect to x, remembering that y is a function of x:

并非所有函数都以显式形式 y = f(x) 给出。隐函数微分用于处理诸如 x² + y² = 25 的方程，其中 y 以隐式定义。对方程两边同时对 x 求导，记住 y 是 x 的函数：

2x + 2y·(dy/dx) = 0 → dy/dx = −x/y

For parametric equations, where x = f(t) and y = g(t), the chain rule provides:

dy/dx = (dy/dt) / (dx/dt)

For example, if x = t² and y = 3t, then dx/dt = 2t and dy/dt = 3, giving dy/dx = 3/(2t). Note that the final answer is expressed in terms of t, not x.

对于参数方程，其中 x = f(t) 且 y = g(t)，链式法则给出：

dy/dx = (dy/dt) / (dx/dt)

例如，若 x = t²，y = 3t，则 dx/dt = 2t，dy/dt = 3，因此 dy/dx = 3/(2t)。注意最终答案以 t 表示，而非 x。

ESAT tip: Implicit differentiation of circles and ellipses is a recurring favourite — memorise that for x² + y² = r², the slope is always −x/y. This saves valuable time.

ESAT技巧：圆和椭圆的隐函数微分是高频考点——记住对于 x² + y² = r²，切线斜率恒为 −x/y。这会为你节省宝贵时间。

8. Second Derivatives and Their Significance | 二阶导数及其意义

The second derivative, denoted f″(x) or d²y/dx², is the derivative of the derivative. It reveals crucial information about the function’s curvature and is essential for classifying stationary points.

二阶导数，记作 f″(x) 或 d²y/dx²，是导数的导数。它揭示函数曲率的关键信息，对于分类驻点至关重要。

Interpretations of the second derivative include:

二阶导数的意义包括：

Concavity: If f″(x) > 0 on an interval, the curve is concave up (shaped like a cup ∪); if f″(x) < 0, the curve is concave down (shaped like a cap ∩).
凹向：若在区间上 f″(x) > 0，曲线凹向上（形如杯 ∪）；若 f″(x) < 0，曲线凹向下（形如帽 ∩）。
Acceleration: In kinematics, d²x/dt² is acceleration — the rate of change of velocity, which itself is dx/dt.
加速度：在运动学中，d²x/dt² 是加速度——即速度（dx/dt）的变化率。
Points of inflection: Where f″(x) changes sign, the curve changes concavity. These points satisfy f″(x) = 0, but this condition is necessary, not sufficient.
拐点：当 f″(x) 改变符号时，曲线改变凹向。拐点满足 f″(x) = 0，但该条件是必要非充分的。

ESAT practice: A typical question gives y = x³ − 6x² + 9x and asks for intervals where the graph is concave up or down. Compute f″(x) = 6x − 12 and solve the inequality — this is a reliable 1-minute question.

ESAT练习：典型题目为 y = x³ − 6x² + 9x，要求判断图像在哪些区间凹向上或凹向下。计算 f″(x) = 6x − 12 并求解不等式——这是一道可靠的一分钟题。

9. Tangents, Normals and Stationary Points | 切线与法线、驻点与最值

At a given point (a, f(a)), the equation of the tangent line is found using the derivative as the slope:

y − f(a) = f′(a)(x − a)

The normal line is perpendicular to the tangent, so its slope is the negative reciprocal: −1/f′(a), provided f′(a) ≠ 0.

在给定点 (a, f(a)) 处，切线方程利用导数作为斜率来求：

y − f(a) = f′(a)(x − a)

法线与切线垂直，因此其斜率为导数的负倒数：−1/f′(a)，前提是 f′(a) ≠ 0。

Stationary points occur where f′(x) = 0. To classify them, use either the first or second derivative test:

First derivative test: Examine the sign of f′(x) just before and after the stationary point. A change from + to − indicates a local maximum; − to + indicates a local minimum; no sign change indicates a point of inflection.
一阶导数检验：考察驻点前后 f′(x) 的符号。从 + 变为 − 表示局部极大值；从 − 变为 + 表示局部极小值；符号不变则为拐点。
Second derivative test: If f″(x) > 0 at the stationary point, it is a local minimum; if f″(x) < 0, a local maximum; if f″(x) = 0, the test is inconclusive.
二阶导数检验：若驻点处 f″(x) > 0，则为局部极小值；若 f″(x) < 0，则为局部极大值；若 f″(x) = 0，则该检验无法判定。

ESAT tip: Questions on stationary points are almost guaranteed in the mathematics section. They often require you to compute the derivative, solve f′(x) = 0, and then apply the second derivative test — a three-step process worth mastering.

ESAT技巧：驻点相关问题在数学部分几乎是必考题。通常需要你求导、解方程 f′(x) = 0、再用二阶导数检验分类——这是一个三步流程，务必熟练掌握。

10. Optimisation and ESAT Exam Strategy | 最优化问题与ESAT应试策略

Optimisation applies differentiation to real-world problems: finding the maximum or minimum value of a quantity subject to constraints. The systematic approach is:

Define the variable(s) and express the quantity to be optimised as a function of one variable.
Differentiate the function and set the derivative equal to zero: f′(x) = 0.
Solve for x and verify the nature (max or min) using the second derivative test or sign analysis.
For closed intervals, also evaluate the function at the endpoints.

最优化问题将微分应用于现实场景：在约束条件下求某个量的最大值或最小值。系统化步骤如下：

定义变量，并将待优化的量表达为单变量函数。
对函数求导并令导数为零：f′(x) = 0。
解方程求 x，并用二阶导数检验或符号分析验证性质（最大值还是最小值）。
对于闭区间问题，还需计算区间端点的函数值。

Classic ESAT-style optimisation questions include: maximising the area of a rectangle with a fixed perimeter, minimising the surface area of a cylinder for a given volume, and finding the maximum height of a projectile from a quadratic position function.

典型的ESAT风格优化题包括：固定周长下使矩形面积最大化、给定体积下使圆柱表面积最小化，以及从二次位置函数求抛射体的最大高度。

ESAT exam strategy: The mathematics component is a multiple-choice test with roughly 50 minutes for 27 questions. Differentiation questions typically alternate between routine computations and conceptual applications. Allocate no more than 1.5–2 minutes per question. If a computation feels convoluted, look for a shortcut — many questions test the product rule or chain rule in isolation, and distractors often reveal the expected method.

ESAT应试策略：数学部分为选择题，约50分钟完成27题。微分题通常在常规计算与概念应用之间交替出现。每题建议控制在1.5–2分钟以内。如果计算过于复杂，寻找捷径——许多题目只单独考查乘积法则或链式法则，干扰项往往能揭示出题者的意图。

Final revision checklist: power rule with fractional and negative indices, product and quotient rules, chain rule fluency, standard derivatives of trigonometric/exponential/logarithmic functions, implicit differentiation, parametric differentiation, second derivative and concavity, tangent/normal equations, stationary point classification, and optimisation setup. Master each block individually, then combine, and differentiation will become one of the most reliable scoring areas in your ESAT paper.

最终复习清单：幂法则（含分数与负指数）、乘积法则与商法则、链式法则熟练度、三角函数/指数/对数函数的标准导数、隐函数微分、参数方程微分、二阶导数与凹凸性、切线与法线方程、驻点分类、优化建模。逐块掌握，再融会贯通，微分将成为你在ESAT考试中最稳定的得分板块之一。

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ESAT Mathematics: Calculus & Differentiation | ESAT数学：微积分与微分专题