📚 Differentiation Dominoes: Core AQA International A Level Practice | 微分多米诺：AQA 国际 A Level 核心练习
This revision article turns AQA International A Level differentiation into a dominoes matching challenge. You will review first principles, the power rule, chain rule, product rule, quotient rule, trigonometric, exponential and logarithmic derivatives, then apply them to a mini worksheet that can be printed and cut into domino cards. Each function must touch its matching derivative, so every domino placed is a small proof of understanding.
这篇复习文章将 AQA 国际 A Level 微分内容变成一个多米诺匹配挑战。你将复习第一原理、幂法则、链式法则、乘积法则、商法则、三角函数、指数与对数函数的导数，然后完成一份可以打印并裁剪成多米诺卡片的迷你练习。每个函数必须与其匹配的导数相接，因此每放下一张多米诺都是对理解的一次小验证。
1. What Are Differentiation Dominoes? | 什么是微分多米诺？
In a differentiation dominoes worksheet, each domino has two ends: one end shows a function and the other end shows a derivative. The cards are shuffled, and the goal is to build a continuous chain in which every function end touches the end showing its derivative. A correct chain forms a closed loop or a specified sequence.
在微分多米诺练习中，每张多米诺有两端：一端是函数，另一端是导数。卡片被洗乱，目标是构建一条连续链，使每个函数端都与其导数端相接。正确的链条会形成一个闭环或指定的顺序。
2. Domino Set-Up and AQA Skill Map | 多米诺设置与 AQA 技能地图
A typical AQA International A Level worksheet includes 12 to 20 dominoes covering AS and A2 topics: powers of x, e to the kx, natural logarithms, sine, cosine, tangent, and the chain, product and quotient rules. Before cutting the cards, label each domino with a number so that you can check a completed loop quickly.
典型的 AQA 国际 A Level 工作表包含 12 到 20 张多米诺，涵盖 AS 和 A2 主题：x 的幂、e 的 kx 次方、自然对数、正弦、余弦、正切，以及链式法则、乘积法则和商法则。裁剪卡片之前，给每张多米诺编号，以便快速检查完成的环路。
3. First Principles and Power Rule Review | 第一原理与幂法则复习
The derivative from first principles is the limit of the gradient of a chord as the interval tends to zero. For any power n, the derivative of x to the n is n times x to the n minus one. This is the core domino pattern for all polynomial cards.
第一原理定义的导数是当间隔趋近于零时弦的斜率的极限。对于任意幂 n，x 的 n 次方的导数是 n 乘以 x 的 n 减一次方。这是所有多项式多米诺卡片的核心模式。
f'(x) = lim (h → 0) [f(x + h) – f(x)] / h
d/dx (xⁿ) = n xⁿ⁻¹
For example, if f(x) = x⁵, then f'(x) = 5x⁴. If f(x) = 4x³, then f'(x) = 12x².
例如，若 f(x) = x⁵，则 f'(x) = 5x⁴。若 f(x) = 4x³，则 f'(x) = 12x²。
4. Chain Rule Dominoes | 链式法则多米诺
The chain rule is used when one function is inside another. Write y as a function of u, and u as a function of x, then multiply the two derivatives. In dominoes, a chain-rule card often has a bracket raised to a power, so the derivative must include the derivative of the inner bracket.
当一个函数嵌套在另一个函数内部时使用链式法则。将 y 写成 u 的函数，u 写成 x 的函数，然后将两个导数相乘。在多米诺中，链式法则卡片通常带有括号的幂，因此导数必须包含内层括号的导数。
dy/dx = dy/du × du/dx
Example: y = (3x² + 5)⁴. Let u = 3x² + 5, so dy/du = 4u³ and du/dx = 6x. Therefore dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.
示例：y = (3x² + 5)⁴。令 u = 3x² + 5，则 dy/du = 4u³，du/dx = 6x。因此 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。
5. Product Rule Dominoes | 乘积法则多米诺
The product rule is needed when two different functions are multiplied. Differentiate the first and leave the second, then add the first left unchanged times the derivative of the second. Domino cards often pair x to a power with sine, cosine or an exponential.
当两个不同的函数相乘时需要使用乘积法则。先对第一个函数求导并保持第二个不变，再加上第一个函数保持不变乘以第二个函数的导数。多米诺卡片常将 x 的幂与正弦、余弦或指数函数配对。
d/dx (uv) = u’v + uv’
Example: y = x² eˣ. Let u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, so dy/dx = 2x eˣ + x² eˣ = x eˣ(x + 2).
示例：y = x² eˣ。令 u = x²，v = eˣ。则 u’ = 2x，v’ = eˣ，所以 dy/dx = 2x eˣ + x² eˣ = x eˣ(x + 2)。
6. Quotient Rule Dominoes | 商法则多米诺
The quotient rule handles a function divided by another function. The denominator times the derivative of the numerator comes first, then subtract the numerator times the derivative of the denominator, all over the denominator squared. The subtraction order must be kept exactly as written.
商法则处理一个函数除以另一个函数的情况。分母乘以分子的导数在前，然后减去分子乘以分母的导数，整体除以分母的平方。减法的顺序必须严格保持。
d/dx (u/v) = (v u’ – u v’) / v²
Example: y = x²/(x + 1). Let u = x² and v = x + 1. Then u’ = 2x and v’ = 1, so dy/dx = [(x + 1)2x – x²(1)] / (x + 1)² = x(x + 2)/(x + 1)².
示例：y = x²/(x + 1)。令 u = x²，v = x + 1。则 u’ = 2x，v’ = 1，所以 dy/dx = [(x + 1)2x – x²(1)] / (x + 1)² = x(x + 2)/(x + 1)²。
7. Exponentials, Logarithms and Trigonometry | 指数、对数与三角函数
These standard derivatives appear constantly in AQA International A Level questions. Learn them as fixed domino ends: the exponential e to the kx brings down the constant k, the derivative of ln x is 1 over x, and the trigonometric derivatives keep their cyclic sign patterns.
这些标准导数在 AQA 国际 A Level 题目中经常出现。将它们作为固定的多米诺端点来记忆：e 的 kx 次方会带下常数 k，ln x 的导数是 1 除以 x，三角函数的导数保持其循环符号规律。
- d/dx (eᵏˣ) = k eᵏˣ
- d/dx (ln x) = 1/x
- d/dx (sin kx) = k cos kx
- d/dx (cos kx) = -k sin kx
- d/dx (tan kx) = k sec² kx
For example, d/dx (sin 4x) = 4 cos 4x and d/dx (cos 2x) = -2 sin 2x. Note that the derivative of cos kx has a negative sign.
例如，d/dx (sin 4x) = 4 cos 4x，d/dx (cos 2x) = -2 sin 2x。注意 cos kx 的导数带有负号。
8. Second Derivatives and Rates of Change | 二阶导数与变化率
The second derivative is the derivative of the derivative. It shows whether the gradient is increasing or decreasing, which links to acceleration in kinematics and to concavity in curve sketching. Domino worksheets can include second derivative cards that require differentiating twice.
二阶导数是导数的导数。它表示斜率是在增大还是在减小，这与运动学中的加速度以及曲线作图中的凹凸性相关。多米诺练习可以包含需要求两次导的二阶导数卡片。
d²y/dx² = d/dx (dy/dx)
Example: s = t³ – 6t² + 9t. Then ds/dt = 3t² – 12t + 9 and d²s/dt² = 6t – 12. At t = 3, d²s/dt² = 6(3) – 12 = 6, so acceleration is positive.
示例：s = t³ – 6t² + 9t。则 ds/dt = 3t² – 12t + 9，d²s/dt² = 6t – 12。当 t = 3 时，d²s/dt² = 6(3) – 12 = 6，因此加速度为正。
9. Tangents, Normals and Stationary Points | 切线、法线与驻点
The derivative gives the gradient of a curve at a point. Use it to find the equation of a tangent or normal. A stationary point occurs where dy/dx equals zero, and its nature can be tested with the second derivative or by inspecting the sign change of dy/dx.
导数给出曲线在某一点的斜率。用它来求切线或法线的方程。驻点出现在 dy/dx 等于零的位置，其性质可以通过二阶导数或检查 dy/dx 的符号变化来判断。
Example: for y = x² + 3 at x = 1, dy/dx = 2x so the gradient is 2. The tangent at (1, 4) is y – 4 = 2(x – 1), and the normal has gradient -1/2, so y – 4 = -½(x – 1).
示例：对于 y = x² + 3 在 x = 1 处，dy/dx = 2x，因此斜率为 2。在点 (1, 4) 处的切线为 y – 4 = 2(x – 1)，法线斜率为 -1/2，所以法线为 y – 4 = -½(x – 1)。
10. Common Errors and How Dominoes Expose Them | 常见错误与多米诺如何暴露它们
Dominoes are useful because a wrong derivative immediately fails to match the next card. The most common AQA errors are forgetting the inner derivative in the chain rule, missing the negative sign when differentiating cos kx, subtracting the quotient rule terms in the wrong order, and applying the product rule to only one factor.
多米诺之所以有用，是因为错误导数会立即无法匹配下一张卡片。AQA 考试中最常见的错误包括：链式法则中忘记内层导数、对 cos kx 求导时漏掉负号、商法则中减法顺序颠倒、以及乘积法则只对其中一个因式求导。
If a chain stops matching, go back to the last placed domino and re-check the derivative. This self-checking structure makes the worksheet ideal for independent revision.
如果链条停止匹配，就回到最后放置的多米诺并重新检查导数。这种自我检查结构使该工作表非常适合独立复习。
11. Full Mini-Worksheet: Match the Dominoes | 完整迷你练习：匹配多米诺
Below are ten function ends and ten derivative ends. The derivative ends have been shuffled. Match each function with its derivative, then arrange them as a domino chain. Write the letter of the matching derivative next to each function number.
下面是十个函数端和十个导数端。导数端已被打乱。将每个函数与其导数匹配，然后把它们排列成多米诺链。在每个函数编号旁写下匹配导数的字母。
| Function end | Shuffled derivative end |
|---|---|
| 1. f(x) = x⁴ | A. 12(2x – 1)⁵ |
| 2. f(x) = e³ˣ | 更多咨询请联系16621398022（同微信）
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