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Differentiation: Rules, Applications and Exam Technique | 微分：法则、应用与考试技巧

2026-08-29 13:45 · 本文部分内容由AI辅助创作

📚 Differentiation: Rules, Applications and Exam Technique | 微分：法则、应用与考试技巧

Differentiation is a core Pure Mathematics topic in Edexcel A-Level. It measures how a function changes as its input changes, giving the gradient of a curve and the instantaneous rate of change. Mastery of the rules and applications of differentiation is essential for both the Pure papers and many applied modelling problems, from kinematics in mechanics to optimisation in statistics and economics contexts.

微分是爱德思 A-Level 纯数学的核心主题。它衡量函数随输入变化而变化的速度，给出曲线的斜率与瞬时变化率。掌握微分法则及其应用，对纯数试卷和许多应用建模问题都至关重要，从力学中的运动学，到统计与经济情境中的最优化问题都离不开它。

This article keeps the original focus of the topic but expands it into a complete revision guide: the definition from first principles, notation, standard derivatives, the product, quotient and chain rules, tangents and normals, stationary points, implicit and parametric differentiation, connected rates of change, and finally exam technique with the errors that cost the most marks.

本文保留原主题的核心内容，并扩写为一篇完整的复习指南：从第一性原理的定义、记号、常用导数，到乘积法则、商法则与链式法则，切线法线、驻点、隐函数与参数微分、相关变化率，最后是考试技巧与最容易丢分的常见错误。

1. What Differentiation Actually Measures | 微分到底在衡量什么

Differentiation is the process of finding the derivative of a function. The derivative measures how the output of a function changes as its input changes, so it tells you the gradient of the curve at a point and the instantaneous rate of change of one quantity with respect to another.

微分就是求一个函数的导数的过程。导数衡量函数输出随输入变化而变化的方式，因此它给出曲线在某一点的斜率，以及一个量相对于另一个量的瞬时变化率。

On a graph, the average rate of change between two points is the gradient of the chord joining them. As the two points move closer together, the chord approaches the tangent, and the average rate of change approaches the instantaneous rate of change. That limiting process is exactly what the derivative formalises.

在图像上，两点之间的平均变化率就是连接这两点的弦的斜率。当两点不断靠近时，弦趋向于切线，平均变化率也趋向于瞬时变化率。导数正是把这个取极限的过程严格化。

Average gradient of chord = [f(x + h) − f(x)] / h

Typical exam language includes "find the rate of change", "find the gradient of the tangent", "show that the curve is increasing", or "find the maximum value". All of these are differentiation tasks in disguise, and recognising the phrasing quickly is half the battle in the exam hall.

常见的考题表述包括"求变化率""求切线的斜率""证明曲线递增"或"求最大值"。这些本质上都是微分任务，在考场上能迅速识别这些措辞，等于赢了一半。

2. First Principles: The Definition of the Derivative | 第一性原理：导数的定义

The derivative of a function f at a point x is defined as the limit of the average gradient from x to x + h as h tends to zero. This gives the gradient of the tangent at that point, and it is the definition from which every rule in the rest of this article is derived.

函数 f 在 x 处的导数，定义为从 x 到 x+h 的平均斜率当 h 趋于 0 时的极限。它给出曲线在该点切线的斜率，也是本文后面所有公式的来源。

f′(x) = lim (h → 0) [f(x + h) − f(x)] / h

To differentiate from first principles, you follow a fixed four-step routine: write down f(x + h), subtract f(x), divide by h and simplify, then let h tend to zero. Edexcel regularly asks for this method, especially for simple polynomials, so the routine must be automatic.

用第一性原理求导要遵循固定的四步：写出 f(x+h)，减去 f(x)，除以 h 并化简，最后令 h 趋于 0。爱德思经常考查这个方法，尤其是对简单多项式，因此这套流程必须做到自动化。

plifies to −1 / [x(x + h)], which tends to −1/x². Both results match the power rule, which is not a coincidence: the power rule is proved exactly this way.

对于 f(x) = x²：f(x+h) − f(x) = (x+h)² − x² = 2xh + h²。除以 h 得 2x + h，令 h → 0 得 f′(x) = 2x。对于 f(x) = 1/x，差商化简为 −1/[x(x+h)]，趋于 −1/x²。两个结果都与幂法则一致，这并非巧合：幂法则正是用这个方法证明的。

3. Notation and Standard Derivatives | 记号与常用导数

. The value of the derivative at x = a is written f′(a) or dy/dx evaluated at that point, and it equals the gradient of the curve there.

导数的记号有多种，必须能自如切换。若 y = f(x)，导数写作 dy/dx、f′(x) 或 y′。在 x = a 处的导数值写作 f′(a)，也就是在该点求 dy/dx，它等于曲线在这一点处的斜率。

The second derivative, written d²y/dx² or f″(x), is the derivative of the derivative. It measures how the gradient itself is changing and is used to determine the nature of stationary points and the concavity of a curve.

二阶导数写作 d²y/dx² 或 f″(x)，是导数的导数。它衡量斜率本身如何变化，用于判断驻点的性质以及曲线的凹凸性。

y or f(x)dy/dx or f′(x)
k (constant)0
xⁿnxⁿ⁻¹
eˣeˣ
ln x1/x
sin xcos x
cos x−sin x
tan xsec²x
sin(kx)k cos(kx)
cos(kx)−k sin(kx)
ekxk ekx

Notice that angles must be in radians whenever trigonometric functions are differentiated. Differentiating sin x with x measured in degrees gives a completely different scaling factor, and using degrees is a classic source of lost marks.

注意：对三角函数求导时，角度必须以弧度为单位。若以度为单位对 sin x 求导，缩放系数完全不同，使用度数是典型的失分原因。

4. The Power Rule with Negative and Fractional Indices | 负指数与分数指数的幂法则

y/dx = nxⁿ⁻¹, and this holds for any rational n, including negative and fractional values.

在对含根式或分式的表达式求导之前，先把它改写为指数形式。幂法则指出，若 y = xⁿ，则 dy/dx = nxⁿ⁻¹，它对任意有理数 n 都成立，包括负数与分数。

So √x = x^½ differentiates to ½x−½, which is 1/(2√x es to −2x⁻³, that is −2/x³. Writing the answer back in the original form is usually expected unless the question says otherwise.

因此 √x = x^½ 求导得 ½x−½，即 1/(2√x)。同理 1/x² = x⁻² 求导得 −2x⁻³，即 −2/x³。除非题目另有要求，通常需要把答案写回原来的形式。

check each term separately; the most common slip is forgetting the sign in front of a negative term.

多项式逐项求导，常数项求导为零。例如 y = 4x³ − 5x² + 7x − 2，则 dy/dx = 12x² − 10x + 7。务必逐项检查，最常见的失误是忘记负项前面的符号。

5. Product, Quotient and Chain Rules | 乘积法则、商法则与链式法则

When a function is built from other functions, the simple power rule is not enough. The product rule handles two functions multiplied together, the quotient rule handles one divided by another, and the chain rule handles a function of a function.

当函数由其他函数组合而成时，单纯用幂法则是不够的。乘积法则处理两个函数相乘，商法则处理两个函数相除，链式法则处理复合函数。

(uv)′ = u′v + uv′

(u/v)′ = (u′v − uv′) / v²

dy/dx = (dy/du) × (du/dx)

(2x + 1)⁴. The chain rule is the single most used rule in the whole A-Level specification.

对于 y = x² sin x，取 u = x²、v = sin x，得 dy/dx = 2x sin x + x² cos x。对于 y = (2x + 1)⁵，令 u = 2x + 1，则 y = u⁵，dy/dx = 5u⁴ × 2 = 10(2x + 1)⁴。链式法则是整个 A-Level 大纲中使用最频繁的法则。

Several practical tips reduce errors. For the chain rule, always identify the inner function first and write u explicitly before differentiating. For the quotient rule, keep the order u′v − uv′ exactly, because reversing the terms flips the sign of the answer. For the product rule, never assume (uv)′ equals u′v′; that mistake appears in examiner reports every single year.

几个实用技巧能减少错误。使用链式法则时，先确定内层函数并明确写出 u 再求导。使用商法则时，严格保持 u′v − uv′ 的顺序，因为颠倒两项会使答案符号相反。使用乘积法则时，切忌认为 (uv)′ 等于 u′v′；这个错误每年都会出现在考官的评述报告中。

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6. Tangent Lines, Normals and Curve Equations | 切线、法线与曲线方程

int (x₁, y₁) is m = f′(x₁), and the tangent line is found using the straight-line formula.

掌握求导之后，求切线与法线就变成常规的代数运算。曲线 y = f(x) 在点 (x₁, y₁) 处切线的斜率是 m = f′(x₁)，切线方程用直线公式求出。

y − y₁ = m(x − x₁)

The normal is the line perpendicular to the tangent at the same point, so its gradient is −1/m. If the tangent is horizontal, the normal is vertical; if the tangent is vertical, the normal is horizontal. Signs matter here more than anywhere else in the topic.

法线是在同一点处与切线垂直的直线，因此其斜率为 −1/m。若切线水平，则法线竖直；若切线竖直，则法线水平。在这个主题中，符号在这里最容易出错。

A very common exam question asks for points on a curve where the tangent is parallel to a given line. Set f′(x) equal to that line's gradient, solve for x, then substitute back into the original equation to obtain the full coordinates. Always give coordinates as ordered pairs, not just x-values.

常见的考题是求曲线上切线平行于已知直线的点。令 f′(x) 等于该直线的斜率，求出 x，再代回原方程得到完整坐标。务必写成有序数对，而不是只写 x 值。

7. Stationary Points and the Second Derivative Test | 驻点与二阶导数判别法

. They include local maxima, local minima and points of inflection with zero gradient, and classifying them correctly is a standard exam requirement.

驻点出现在斜率为零的地方，即 f′(x) = 0。它包括局部极大值、局部极小值以及斜率为零的拐点，正确判断其类型是标准考试要求。

) = 0, the test is inconclusive and you must check the sign of f′(x) on either side of the point.

二阶导数判别法如下：在驻点处若 f″(x) < 0，为局部极大值；若 f″(x) > 0，为局部极小值；若 f″(x) = 0，判别法失效，必须检查该点两侧 f′(x) 的符号。

al maximum; at x = 2 it equals 6, a local minimum. Substituting back gives the y-coordinates and therefore the full points.

对于 y = x³ − 3x² + 4，dy/dx = 3x² − 6x = 3x(x − 2)，故驻点在 x = 0 与 x = 2。二阶导数为 d²y/dx² = 6x − 6。在 x = 0 时为 −6，是局部极大值；在 x = 2 时为 6，是局部极小值。代回原式可得 y 坐标，从而得到完整坐标。

of inflection with zero gradient is stationary yet not a turning point. If a question says "show that the curve has no stationary points" u must show that f′(x) = 0 has no real solutions, usually by considering the discriminant.

不要把驻点与转折点混为一谈。所有转折点都是驻点，但斜率为零的拐点是驻点却不是转折点。若题目要求"证明曲线没有驻点"，就要证明 f′(x) = 0 无实根，通常通过判别式来说明。

8. Increasing, Decreasing Functions and Curve Sketching | 增减函数与图像绘制

A function is increasing on an interval where f′(x) > 0 and decreasing where f′(x) < 0. These facts let you prove monotonicity over a given range and are frequently combined with inequalities in exam questions.

函数在 f′(x) > 0 的区间上递增，在 f′(x) < 0 的区间上递减。这些事实可用于证明给定区间上的单调性，在考题中常与不等式结合出现。

ehaviour as x → ±∞, and note any asymptotes. For a cubic with a positive leading coefficient, the shape is a maximum followed by a minimum, or vice versa depending on the roots of the derivative.

掌握导数后，图像绘制会显著提升。先找截距，再求驻点及其性质，检查 x → ±∞ 时的趋势，并标注渐近线。对于首项系数为正的三次函数，图像形状是先极大后极小；若导数根的排列相反，则先极小后极大。

The second derivative also tells you about concavity. Where f″(x) > 0 the curve is concave upwards and any stationary point there is a minimum; where f″(x) < 0 it is concave downwards and a stationary point there is a maximum. This gives a quick sanity check on your answers during the exam.

二阶导数还能说明凹凸性。f″(x) > 0 处曲线开口向上（上凹），该处驻点为极小值；f″(x) < 0 处曲线开口向下（下凹），该处驻点为极大值。这可以在考试中快速检验答案是否合理。

9. Implicit and Parametric Differentiation | 隐函数微分与参数微分

x). In implicit differentiation you differentiate both sides of an equation with respect to x, remembering that y is a function of x, so every term containing y produces a dy/dx factor through the chain rule.

并非所有曲线都能写成 y = f(x)。在隐函数微分中，对等式两边关于 x 求导，记住 y 是 x 的函数，因此每一含 y 的项都会通过链式法则产生一个 dy/dx 因子。

metry: the tangent to a circle is perpendicular to the radius, since their gradients multiply to −1.

对于圆 x² + y² = 25，求导得 2x + 2y(dy/dx) = 0，故 dy/dx = −x/y。这一结果印证了其几何意义：圆的切线与半径垂直，因为二者斜率之积为 −1。

For parametric curves, where x and y are each given in terms of a parameter t, the chain rule gives a compact formula. Differentiate each coordinate with respect to t, then divide.

对于参数曲线，x 与 y 各自以参数 t 表示，链式法则给出一个简洁公式。先分别对 t 求导，再相除即可。

dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0

ifying these special points is often worth a mark on its own, and they are essential when sketching parametric curves.

dx/dt = 0 的点处切线竖直，dy/dt = 0 的点处切线水平。识别这些特殊点往往单独就有一分，而且在绘制参数曲线时必不可少。

10. Connected Rates of Change, Optimisation and Modelling | 相关变化率、最优化与建模

Applied questions use the chain rule to link rates. If a quantity V depends on r, and r depends on time t, then the rates are connected by multiplication, which is the standard tool for related-rates problems such as inflating balloons or filling tanks.

应用题用链式法则把变化率联系起来。若量 V 依赖于 r，而 r 又依赖于时间 t，则各变化率通过相乘相连，这是处理充气球、注水等变化率问题的标准工具。

dV/dt = (dV/dr) × (dr/dt)

In kinematics, differentiation links displacement, velocity and acceleration directly: velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity. The sign of the derivative tells you the direction of motion, which is why a negative velocity does not simply mean "slowing down".

在运动学中，微分直接联系位移、速度与加速度：速度是位移对时间的导数，加速度是速度的导数。导数的符号指示运动方向，这也是为什么速度为负并不等于"减速"。

Optimisation problems follow a fixed pattern: build a formula for the quantity to be maximised or minimised, use any constraint to reduce it to a single variable, differentiate, set the derivative to zero, solve, and then justify that your answer is indeed a maximum or minimum using the second derivative. Always re-read the question to check whether it wants the value of the variable or the maximum value of the quantity itself.

最优化问题遵循固定套路：先建立待求最大值或最小值的量的表达式，利用约束条件化为一元函数，求导，令导数为零，求解，再用二阶导数验证确为极大或极小。最后重读题目，确认它要的是变量的值还是该量的最值。

11. Exam Technique, Common Errors and a Revision Plan | 考试技巧、常见错误与复习计划

Most marks in differentiation questions are lost through algebra rather than through misunderstanding calculus. Expanding brackets before differentiating a product, missing a negative sign, or forgetting the 1/x derivative are among the most frequent slips reported by examiners.

微分题的分大多丢在代数上，而不是理解上。在求乘积的导数前先展开括号、漏掉负号、忘记 1/x 的导数，是考官报告中最常提到的失误。