📚 pdfjoiner_(4)-077: A-Level Differentiation Techniques | pdfjoiner_(4)-077: A-Level 微分技巧
)-077 brings together the core differentiation techniques required for Edexcel A-Level Mathematics. Differentiation underpins everything from finding gradients of curves to solving real-world rate-of-change problems. Mastering these methods will give you a solid foundation for Paper 1 and Paper 2, especially when tackling optimisation, curve sketching, and connected rates questions.
本份 pdfjoiner_(4)-077 合并笔记汇集了 Edexcel A-Level 数学中所有核心的微分技巧。微分是贯穿整个课程的基础工具，无论是求曲线斜率、求解最值问题还是应用相关变化率，都离不开它。熟练掌握这些方法，你就能在 Paper 1 与 Paper 2 中从容应对优化、曲线描绘以及速率关联等题型。
1. Gradient Function and the Limit Definition | 梯度函数与极限定义
Differentiation finds the gradient of a curve y = f(x) at any point. The derivative f'(x) is defined as the limit of the difference quotient: f'(x) = limₕ→₀ [f(x+h) – f(x)] / h. Although you won' ly, understanding it helps connect average rate of change to instantaneous rate of change.
微分用来求曲线 y = f(x) 上任意一点处的梯度。导数 f'(x) 定义为差商的极限：f'(x) = limₕ→₀ [f(x+h) – f(x)] / h。虽然你很少直接使用这个定义，但理解它有助于把平均变化率和瞬时变化率联系起来。
imple functions like x². Practice writing the expression, expanding, cancelling h, and letting h → 0.
A-Level 更注重规则的运用而非极限的计算。不过，考试可能会要求你对 x² 这类简单函数进行第一原理求导。练习写出表达式、展开、约去 h 并令 h → 0。
2. Power Rule and Constant Multiple Rule | 幂法则与常数倍法则
If f(x) = xⁿ, then f' king it extremely versatile. For instance, d/dx (x⁵) = 5x⁴, d/dx (x⁻²) = -2x⁻³, and d/dx (√x) = d/dx (x½) = ½x−½.
若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹，其中 n 为实数。这个法则适用于正整数、负整数以及分数指数，用途极广。例如，d/dx (x⁵) = 5x⁴，d/dx (x⁻²) = -2x⁻³，d/dx (√x) = d/dx (x½) = ½x−½。
The constant multiple rule says that d/dx [k f(x)] = k f'(x). Together with the sum rule, d/dx [f(x) ± g(x)] = f'(x) ± g'(x), you can differentiate any polynomial term-by-term. Example: y = 4x³ – 7x + 2 gives dy/dx = 12x² – 7.
常数倍法则指出 d/dx [k f(x)] = k f'(x)。再结合和差法则 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)，你就能逐项对任何多项式求导。例如 y = 4x³ – 7x + 2 得到 dy/dx = 12x² – 7。
3. Chain Rule - Function of a Function | 链式法则——复合函数求导
When y = f(g(x)), the chain rule states that dy/dx = f'(g(x)) × g' = (dy/du) × (du/dx). This is essential for differentiating expressions like (3x² + 5)⁴, e2x, or ln(5x – 1).
当 y = f(g(x)) 时，链式法则为 dy/dx = f'(g(x)) × g'(x)。用莱布尼茨记号表示：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。这对求导 (3x² + 5)⁴、e2x 或 ln(5x – 1) 等表达式至关重要。
Step-by-step: let u be the inner function, differentiate u with respect to x, differentiate y with respect to u, then multiply. Always remember to substitute back so the final derivative is expressed in terms of x.
步骤分解：令 u 为内层函数，求 u 对 x 的导数，再求 y 对 u 的导数，最后相乘。务必记得回代，使最终导数用 x 表示。
d/dx [(2x + 1)³] = 3(2x + 1)² × 2 = 6(2x + 1)²
4. Product Rule | 乘积法则
For y = u(x) v(x), the derivative is dy/dx = u (dv/dx) + v (du/dx). It's helpful to write 'first times derivative of second plus second times derivative of first' dexcel often tests this with trigonometric, exponential, and algebraic functions combined.
若 y = u(x) v(x)，则导数为 dy/dx = u (dv/dx) + v (du/dx)。记忆技巧为“前微后不微，加后微前不微”。Edexcel 常将它与三角函数、指数函数和代数函数混合在一起考查。
Example: differentiate y = x² sin x. Let u = x², v = sin x. Then u' = 2x, v' e: dy/dx = x² cos x + 2x sin x. Always factorise if possible to leave a tidy final answer.
例题：对 y = x² sin x 求导。设 u = x²，v = sin x。则 u' = 2x，v' = cos x。代入公式：dy/dx = x² cos x + 2x sin x。最后尽量因式分解，使答案整洁。
5. Quotient Rule | 商法则
If y = u(x) / v(x), then dy/dx = [v (du/dx) – u (dv/dx)] / v². Pay close attention to the minus sign in the numerator — getting the order wrong is one of the most common errors in A-Level differentiation.
若 y = u(x) / v(x)，则 dy/dx = [v (du/dx) – u (dv/dx)] / v²。务必注意分子中的减号——搞错顺序是 A-Level 微分中最常见的失误之一。
For y = (3x + 1) / (x – 2), let u = 3x + 1, v = x – 2. Then u' = 3, v' = 1. dy/dx = [(x – 2)(3) – (3x + 1)(1)] / (x – 2)² = (3x – 6 – 3x – 1)/(x – 2)² = -7/(x – 2)². Simplify before writing the final answer.
例如 y = (3x + 1) / (x – 2)，设 u = 3x + 1，v = x – 2。则 u' = 3，v' = 1。dy/dx = [(x – 2)(3) – (3x + 1)(1)] / (x – 2)² = (3x – 6 – 3x – 1)/(x – 2)² = -7/(x – 2)²。务必化简后才写出最终答案。
6. Implicit Differentiation | 隐函数微分
When y is not given explicitly as a function of x, differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule. Every time you differentiate a term in y, multiply by dy/dx. This allows you to find gradients of curves like circles, ellipses, and more.
当 y 未用 x 显式表示时，对方程两边同时对 x 求导，将 y 视为 x 的函数并应用链式法则。每当你对含 y 的项求导时，都要乘上 dy/dx。这使你能求出圆、椭圆等曲线的斜率。
radient at (3, 4) plug in values to get -3/4. Always check that the point satisfies the original equation.
对 x² + y² = 25 求导：2x + 2y (dy/dx) = 0 → dy/dx = -x/y。接着将 (3, 4) 代入得斜率 -3/4。始终要检验该点是否满足原方程。
7. Connected Rates of Change | 相关变化率
) × (dr/dt). This is especially common in questions about filling containers, expanding spheres, or moving ladders.
利用链式法则，你可以将两个或多个变化率关联起来。例如，若体积 V 依赖于半径 r，而 r 又依赖于时间 t，则有 dV/dt = (dV/dr) × (dr/dt)。这在涉及注水、球体膨胀或移动梯子的问题中尤为常见。
lume when r = 10 cm. V = 4/3 π r³ → dV/dr = 4π r². Then dV/dt = 4π (10)² × 2 = 800π cm³/s.
典型例题：一只球形气球以 2 cm/s 的速率充气膨胀。求当 r = 10 cm 时体积的增加率。V = 4/3 π r³ → dV/dr = 4π r²。则 dV/dt = 4π (10)² × 2 = 800π cm³/s。
8. Derivatives of Trigonometric, Exponential & Log Functions | 三角、指数与对数函数的导数
These are standard results you must commit to memory:
- d/dx (sin x) = cos x
- d/dx (cos x) = –sin x
- d/dx (tan x) = sec² x
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x, for x > 0
这些都是你必须牢记的标准结果：
- d/dx (sin x) = cos x
- d/dx (cos x) = –sin x
- d/dx (tan x) = sec² x
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x，x > 0
linear functions, apply the chain rule. For instance, d/dx [sin(3x)] = 3 cos(3x); d/dx [e5x] = 5 e5x; d/dx [ln(2x+1)] = 2/(2x+1). Recognising these patterns speeds up your working.
当这些函数与线性函数复合时，要应用链式法则。例如 d/dx [sin(3x)] = 3 cos(3x)；d/dx [e5x] = 5 e5x；d/dx [ln(2x+1)] = 2/(2x+1)。记住这些模式能提升解题速度。
9. Second Derivative and Stationary Points | 二阶导数与驻点
The second derivative f''(x) measures the rate of change of the gradient. If f''(x) > 0, the curve is convex (∪); if f''(x) < 0, it is concave (∩). Points where f'(x) = 0 are stationary points — they can be maxima, minima, or points of inflection.
二阶导数 f''(x) 衡量切线的变化率。若 f''(x) > 0，曲线呈下凸 (∪)；若 f''(x) < 0，曲线呈上凸 (∩)。满足 f'(x) = 0 的点为驻点——可能是极大值点、极小值点或拐点。
Use the second derivative test: after finding where f'(x) = 0, substitute these x-values into f''(x). If f''(x) < 0, the point is a local maximum; if f''(x) > 0, it's a local minimum. If f''(x) = 0, use the first derivative test or examine sign changes.
运用二阶导数检验：找到 f'(x) = 0 后，将这些 x 值代入 f''(x)。若 f''(x) < 0，该点为极大值点；若 f''(x) > 0，该点为极小值点。若 f''(x) = 0，则使用一阶导数检验或考察符号变化。
10. Practical Tips for Edexcel Differentiation Questions | Edexcel 微分题的实用技巧
Always simplify expressions before differentiating — rewrite roots as fractional powers and bring denominators up as negative powers. For example, write 1/x² as x⁻² and √x³ as x3/2. This allows you to use the power rule efficiently.
求导前务必先化简表达式——将根式写成分数指数，将分母变成负指数。例如把 1/x² 写成 x⁻²，把 √x³ 写成 x3/2。这样就能高效地使用幂法则。
of a tangent or normal, first find dy/dx at the given point for the gradient m. Tangent: y – y₁ = m(x – x₁); normal: gradient = – ur differentiation steps clearly to secure method marks even if arithmetic mistakes occur.
若题目要求求切线或法线方程，先求出给定点处的 dy/dx 作为梯度 m。切线：y – y₁ = m(x – x₁)；法线：梯度 = –1/m（前提 m ≠ 0）。清晰展示求导步骤，即使出现计算错误也能保住方法分。
In applied contexts, always define variables and state which rate you are finding. Write down the chain rule relationship before plugging in numbers. Units matter: velocity is rate of change of displacement; acceleration is rate of change of velocity.
在应用题中，务必定义变量，并说明你要求的是哪个速率。先写出链式法则关系式再代入数值。单位很关键：速度是位移的变化率；加速度是速度的变化率。
11. Common Mistakes to Avoid | 应避免的常见错误
ssic pitfall. Remember sin² x = (sin x)², so its derivative is 2 sin x cos x. Similarly, e-x differentiates to –e-x; forgetting the negative sign loses marks.
像 sin² x 这样的函数求导时遗漏链式法则是典型的陷阱。请记住 sin² x = (sin x)²，因此其导数为 2 sin x cos x。类似地，e-x 的导数为 –e-x；忘记负号就会丢分。
Do not treat product and quotient rules as optional — e.g. writing x²/x = x) is valid only when simplification eliminates the need for the rule.
不要将乘积法则和商法则视为可有可无——看到乘积就应用乘积法则；看到分式就应用商法则。只有通过化简消去分式时（例如将 x²/x 简化为 x）才能绕开它们。
12. Revision Checklist for pdfjoiner_(4)-077 | pdfjoiner_(4)-077 复习清单
| Topic | Key Skills |
| Basic differentiation | Power rule, constant multiple, sum/difference |
| Chain rule | Identify inner/outer functions, multiply by du/dx |
| Product rule | u'v + uv' with correct factors |
| Quotient rule | (vu' – uv')/v², watch the minus sign |
| Implicit differentiation | d/dx of y terms includes dy/dx factor |
| Connected rates | Chain rule dV/dt = dV/dr × dr/dt |
| Second derivative | Nature of stationary points, convex/concave |
| Tangents & normals | Find m from dy/dx, use y – y₁ = m(x – x₁) |
This table summarises the core differentiation topics covered in these combined notes. Use it alongside past papers to track your progress.
本表格汇总了这本合并笔记涵盖的所有核心微分专题。请结合历年真题使用，追踪自己的掌握进度。
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