📚 A-Level Maths Differentiation Dominoes Worksheet | 微分多米诺骨牌活动知识点精讲
Differentiation dominoes is a brilliant hands-on activity that turns a daunting list of derivative rules into an engaging chain of matches. Each domino features a function on one half and a derivative on the other, requiring you to pair them up by applying the correct differentiation technique. This article walks you through every core differentiation topic covered by typical A-Level domino sets, blending clear explanations with practical examples so you can master calculus while playing.
微分多米诺骨牌是一项出色的动手活动，它将一长串令人头疼的导数规则转变为一条条趣味十足的匹配链。每张骨牌一半写着函数，另一半写着导数，你需要运用正确的微分技巧将它们配对起来。这篇文章会带你逐一梳理典型 A-Level 多米诺题目中涉及的所有核心微分知识点，把清晰的讲解和实例结合起来，让你在游戏中掌握微积分。
1. What Are Differentiation Dominoes? | 什么是微分多米诺骨牌？
A differentiation domino set consists of tiles where each tile’s left side shows a derivative expression and its right side shows a different function. The goal is to build a continuous chain by matching a function to its correct derivative found on another tile. This forces you to recognise standard derivatives, apply rules, and check your work actively rather than passively reading notes.
微分多米诺骨牌由一组牌片组成，每张牌左侧显示一个导数表达式，右侧显示另一个函数。游戏目标是通过将函数与其正确的导数匹配来建造一条连续的链条。这个过程迫使你主动识别标准导数、运用求导法则并核验答案，而不是被动地翻阅笔记。
The worksheet version usually provides a scrambled set of tiles that you cut out and rearrange. It might cover basic power rule, chain rule, product and quotient rules, as well as trigonometric, exponential and logarithmic functions. Dominoes are excellent for peer discussion and can highlight common pitfalls, like forgetting the chain rule or confusing a product with a composition.
练习纸版本通常会提供一组打乱的牌片，由你裁剪后重新排列。内容可能涵盖基本幂法则、链式法则、乘积法则和商法则，还有三角、指数和对数函数。多米诺活动非常适合小组讨论，能暴露常见错误，比如忘记链式法则或者混淆乘积与复合函数。
2. Differentiation from First Principles | 从第一原理出发理解导数
Before diving into domino patterns, it helps to recall the definition of the derivative. For a function f(x), the derivative f'(x) is given by the limit: f'(x) = lim(h→0) [f(x+h) – f(x)] / h. This limit measures the instantaneous rate of change and underpins every rule you will apply on the dominoes.
在深入多米诺模式之前，回顾导数的定义会很有帮助。对于函数 f(x)，导数 f'(x) 由极限定义：f'(x) = lim(h→0) [f(x+h) – f(x)] / h。这个极限衡量了瞬时变化率，是整个多米诺活动中你将会用到的每条法则的基石。
For a simple power function like f(x) = x², applying the definition yields f'(x) = 2x. Multiplying out (x+h)² = x² + 2xh + h², subtracting x² gives 2xh + h², dividing by h gives 2x + h, and letting h → 0 leaves 2x. This illustrates why the power rule works and why constants disappear.
对于像 f(x) = x² 这样的简单幂函数，使用定义可得 f'(x) = 2x。展开 (x+h)² = x² + 2xh + h²，减去 x² 得 2xh + h²，除以 h 得 2x + h，令 h → 0 即得 2x。这解释了幂法则的由来以及常数项为何会消失。
3. Power Rule and Constant Multiple Rule | 幂法则与常数倍法则
The power rule states that if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This is the workhorse for polynomial functions and appears frequently on domino tiles. Remember that the exponent n can be any real number, including fractions and negatives.
幂法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。这是处理多项式函数的主力法则，频繁出现在多米诺牌面上。记住指数 n 可以是任何实数，包括分数和负数。
Combined with the constant multiple rule, d/dx [c·f(x)] = c·f'(x), you can differentiate terms like 5x³ directly: bring down the exponent to get 3·5x² = 15x². Domino activities will test your ability to quickly spot the difference between 4x² and x⁴, or to recognise that the derivative of ½x⁻² is -x⁻³.
结合常数倍法则 d/dx [c·f(x)] = c·f'(x)，你可以直接对 5x³ 这样的项求导：把指数拿下来，得到 3·5x² = 15x²。多米诺活动会考查你快速区分 4x² 和 x⁴ 的能力，或者识别出 ½x⁻² 的导数是 -x⁻³。
d/dx (xⁿ) = n xⁿ⁻¹ and d/dx (c·f(x)) = c·f'(x)
4. The Chain Rule | 链式法则
The chain rule is essential when a function is composed of an outer and an inner function: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). In domino sets, this often appears with expressions like sin(3x), (2x+1)⁴, or e^(5x). You must differentiate the outer function, keep the inner unchanged, and multiply by the derivative of the inner.
当一个函数由外层和内层函数复合而成时，链式法则是必不可少的：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。在多米诺牌组中，这常常体现在 sin(3x)、(2x+1)⁴ 或 e^(5x) 等表达式中。你需要先求外层函数的导数，保持内层不变，再乘以内层函数的导数。
For example, to differentiate h(x) = (3x² – 5)⁷, set u = 3x² – 5. Then h(x) = u⁷, so h'(x) = 7u⁶ · u’ = 7(3x² – 5)⁶ · 6x = 42x(3x² – 5)⁶. A common domino mistake is to write 7(3x² – 5)⁶ and stop, forgetting the crucial ‘· 6x’ part.
例如，对 h(x) = (3x² – 5)⁷ 求导，设 u = 3x² – 5。则 h(x) = u⁷，于是 h'(x) = 7u⁶ · u’ = 7(3x² – 5)⁶ · 6x = 42x(3x² – 5)⁶。一个常见的多米诺错误是写出 7(3x² – 5)⁶ 就停笔，忘记了关键的 ‘· 6x’ 部分。
dy/dx = dy/du · du/dx
5. Product Rule | 乘积法则
When a function is the product of two simpler functions, y = u(x)·v(x), we use the product rule: dy/dx = u’v + uv’. Domino tiles often mix products with powers and trig functions, such as x²·sin x or eˣ·ln x. The key is to identify u and v clearly before differentiating.
当一个函数是两个较简单函数的乘积时，y = u(x)·v(x)，我们使用乘积法则：dy/dx = u’v + uv’。多米诺牌面通常会将乘积与幂函数、三角函数混合，比如 x²·sin x 或 eˣ·ln x。关键是在求导前先明确划分 u 和 v。
Take f(x) = (3x – 1)(x² + 2). Let u = 3x – 1, v = x² + 2. Then u’ = 3, v’ = 2x. Applying the product rule: f'(x) = 3·(x² + 2) + (3x – 1)·2x = 3x² + 6 + 6x² – 2x = 9x² – 2x + 6. In dominoes, you might be asked to match this derivative with the expanded form of the original function.
以 f(x) = (3x – 1)(x² + 2) 为例。令 u = 3x – 1, v = x² + 2。则 u’ = 3, v’ = 2x。运用乘积法则：f'(x) = 3·(x² + 2) + (3x – 1)·2x = 3x² + 6 + 6x² – 2x = 9x² – 2x + 6。在多米诺游戏中，你可能需要将这个导数与展开后的原函数进行匹配。
d/dx [u(x)·v(x)] = u'(x)·v(x) + u(x)·v'(x)
6. Quotient Rule | 商法则
For functions written as a fraction, y = u(x)/v(x), the quotient rule gives dy/dx = (u’v – uv’) / v². This often appears in domino sets with rational functions like (x+1)/(x-2) or trigonometric ratios such as tan x = sin x / cos x. Memorising the order ‘low d-high minus high d-low over the square of what’s below’ helps avoid sign errors.
对于写成分式形式的函数 y = u(x)/v(x)，商法则给出 dy/dx = (u’v – uv’) / v²。这一法则在多米诺牌组中常伴随有理函数出现，如 (x+1)/(x-2)，或三角比，如 tan x = sin x / cos x。记住口诀“下面乘上面的导数减去上面乘下面的导数，除以下面的平方”有助于避免符号错误。
Consider y = (2x+1)/(x-3). Here u = 2x+1, v = x-3, so u’ = 2, v’ = 1. Then dy/dx = [2(x-3) – (2x+1)(1)] / (x-3)² = (2x-6 – 2x – 1) / (x-3)² = -7 / (x-3)². When tackling dominoes, always simplify the numerator; a match might hide as -7(x-3)⁻².
考虑 y = (2x+1)/(x-3)。这里 u = 2x+1, v = x-3，因此 u’ = 2, v’ = 1。那么 dy/dx = [2(x-3) – (2x+1)(1)] / (x-3)² = (2x-6 – 2x – 1) / (x-3)² = -7 / (x-3)²。在处理多米诺骨牌时，记得化简分子；配对目标可能以 -7(x-3)⁻² 的形式隐藏。
d/dx [u(x)/v(x)] = [u'(x)v(x) – u(x)v'(x)] / [v(x)]²
7. Derivatives of Trigonometric Functions | 三角函数的导数
Standard A-Level domino sets expect you to know the derivatives of sin x, cos x and tan x off by heart: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, and d/dx (tan x) = sec² x. These must be used seamlessly with the chain rule, as in sin(2x) becoming 2·cos(2x).
标准的 A-Level 多米诺游戏要求你牢记 sin x、cos x 和 tan x 的导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。这些公式必须与链式法则无缝结合，例如 sin(2x) 的导数为 2·cos(2x)。
Reciprocal trig functions also appear: d/dx (sec x) = sec x tan x, d/dx (csc x) = -csc x cot x, d/dx (cot x) = -csc² x. Although less common, they can form tricky domino ends. A useful warm-up is matching the pattern cos³ x, which requires chain rule to give -3 cos² x sin x.
倒数三角函数也会出现：d/dx (sec x) = sec x tan x，d/dx (csc x) = -csc x cot x，d/dx (cot x) = -csc² x。尽管较少见，它们可能构成棘手的多米诺尾端。一个有用的热身是匹配 cos³ x 的模式，它需要链式法则得出 -3 cos² x sin x。
8. Exponential and Logarithmic Differentiation | 指数与对数函数的微分
The derivative of eˣ is remarkably simple: d/dx (eˣ) = eˣ. Combined with the chain rule, d/dx (e^(kx)) = k e^(kx). Domino cards frequently test this with functions like e^(3x+2), where the derivative is 3e^(3x+2). Always watch for the coefficient multiplying the exponent!
eˣ 的导数出奇简单：d/dx (eˣ) = eˣ。结合链式法则，d/dx (e^(kx)) = k e^(kx)。多米诺卡片常以 e^(3x+2) 这类函数检验此点，其导数为 3e^(3x+2)。千万注意指数前面乘的系数！
For natural logarithms, d/dx (ln x) = 1/x for x > 0. More generally, d/dx (ln(f(x))) = f'(x)/f(x). So the derivative of ln(5x) is 5/(5x) = 1/x. Dominoes often exploit this to see if you recognise that ln(5x) and ln(x)+C share the same derivative, reinforcing the idea of integration constants.
对于自然对数，当 x > 0 时，d/dx (ln x) = 1/x。更一般地，d/dx (ln(f(x))) = f'(x)/f(x)。因此 ln(5x) 的导数是 5/(5x) = 1/x。多米诺活动常利用这一点，看你能否识别出 ln(5x) 与 ln(x)+C 具有相同的导数，从而强化积分常数的概念。
9. Implicit Differentiation | 隐函数求导
Implicit differentiation deals with equations where y is not (easily) isolated, like x² + y² = 25. You differentiate both sides with respect to x, treating y as a function of x and applying the chain rule: d/dx (y²) = 2y · dy/dx. After differentiating, solve for dy/dx.
隐函数求导处理的是 y 没有（轻易）被分离出来的方程，如 x² + y² = 25。你对等式两边关于 x 求导，将 y 视为 x 的函数并运用链式法则：d/dx (y²) = 2y · dy/dx。求导后，解出 dy/dx。
For x² + y² = 25, differentiation gives 2x + 2y dy/dx = 0, so dy/dx = -x/y. In a domino context, you might match the implicit form with its explicit derivative or be asked to find the gradient at a given point. The challenge is remembering to multiply by dy/dx whenever you differentiate a y-term.
对于 x² + y² = 25，求导得 2x + 2y dy/dx = 0，于是 dy/dx = -x/y。在多米诺情境中，你可能需要将隐式形式与其显式导数配对，或者求某一点处的梯度。挑战在于，每次对含有 y 的项求导时都要记得乘上 dy/dx。
d/dx (yⁿ) = n yⁿ⁻¹ · dy/dx
10. Higher Order Derivatives and Rates of Change | 高阶导数与变化率
Domino sequences sometimes extend to second derivatives, f”(x) or d²y/dx². After finding f'(x), you simply differentiate again. For kinematics problems, displacement s(t), velocity v(t) = ds/dt, and acceleration a(t) = d²s/dt² form a natural domino trio.
多米诺序列有时会延伸到二阶导数 f”(x) 或 d²y/dx²。在求出 f'(x) 后，只需再求导一次。对于运动学问题，位移 s(t)、速度 v(t) = ds/dt 和加速度 a(t) = d²s/dt² 构成了一个天然的多米诺三重奏。
If f(x) = 4x³ – 2x, then f'(x) = 12x² – 2 and f”(x) = 24x. In a practical worksheet, you might see a tile with ’24x’ on one side and the description ‘acceleration of particle with position function 4x³ – 2x’ on the other. This contextualises pure differentiation and strengthens your understanding of derivatives as rates of change.
如果 f(x) = 4x³ – 2x，那么 f'(x) = 12x² – 2，而 f”(x) = 24x。在实际练习中，你可能会看到一张牌，一面写着 ’24x’，另一面描述为“位置函数为 4x³ – 2x 的粒子的加速度”。这样将纯微分与情境联系起来，能加深你对导数作为变化率的理解。
11. Common Domino Traps and Tips | 多米诺常见陷阱与技巧
One of the biggest trap in differentiation dominoes is mixing up the chain rule with the product rule. When seeing x·sin(2x), many students wrongly write cos(2x) or 2x·cos(2x). The correct derivative requires the product rule: derivative of x is 1, so 1·sin(2x) + x·(cos(2x)·2) = sin(2x) + 2x·cos(2x).
微分多米诺最大的陷阱之一是把链式法则和乘积法则搞混。看到 x·sin(2x) 时，许多学生错误地写成 cos(2x) 或 2x·cos(2x)。正确的导数需要乘积法则：x 的导数是 1，所以 1·sin(2x) + x·(cos(2x)·2) = sin(2x) + 2x·cos(2x)。
Another recurring error is mishandling negative signs, particularly with cos x and in the quotient rule. Before starting the domino activity, it is wise to write down a quick formula sheet and keep it visible. Match the easy linear derivatives first, then tackle the compositions and products. If a domino seems stuck, differentiate both possible functions on a scrap of paper to verify.
另一个常见错误是处理负号不当，尤其涉及 cos x 和商法则时。开始多米诺活动前，明智的做法是先写一张简短的公式表并放在显眼处。先匹配简单的线性导数，再处理复合与乘积。如果某张牌卡住了，不妨在草稿纸上对两边可能的函数都求一下导来验证。
12. Building Your Own Domino Set for Revision | 自制多米诺牌组复习
Creating your own differentiation dominoes is one of the most effective revision techniques. Choose 12–15 functions that cover a mix of rules. On the left of each card write a derivative, and on the right write an unrelated function. Your set should form a closed loop or a single linear chain, ensuring that every derivative matches exactly one function in the set.
自制微分多米诺骨牌是最有效的复习技巧之一。挑选 12–15 个涵盖各种法则的函数。在每张卡的左侧写上一个导数，右侧写上一个不相关的函数。你的牌组应当形成一个闭合的环或单一线性链，确保每个导数恰好匹配组中的一个函数。
Swap your set with a classmate and time yourselves assembling the chain. Discuss any mismatches and correct them together. The tactile nature of moving tiles reinforces memory far better than simply reading a list of derivatives. You can even colour-code tiles by rule type – green for chain, blue for product – to create a visual map of your differentiation toolkit.
和同学交换牌组，并计时完成链条的拼接。讨论任何不匹配的地方并一起纠正。移动牌片的动手特性比单纯阅读导数列表更能强化记忆。你甚至可以按法则类型为牌片涂上颜色——绿色代表链式，蓝色代表乘积——为你的微分工具箱创建一张视觉地图。
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