📚 Edexcel A-Level Maths: Differentiation and Applications | Edexcel A-Level 数学：微分及其应用
Differentiation is one of the most important topics in Edexcel A-Level Mathematics. It measures instantaneous rates of change and underpins optimisation, kinematics and curve sketching, so you will meet it in Paper 1 and Paper 2.
微分是 Edexcel A-Level 数学最重要的主题之一。它度量瞬时变化率，是优化、运动学和曲线草图的基础，因此你会在 Paper 1 和 Paper 2 中遇到。
1. The Meaning of Differentiation | 微分的意义
Differentiation gives the gradient of a curve at a particular point. If y = f(x), the derivative dy/dx is the rate at which y changes with respect to x.
微分给出曲线上某一点处的梯度。如果 y = f(x)，导数 dy/dx 表示 y 随 x 变化的速率。
The derivative is also written as f'(x) or d/dx[f(x)]. At a point x = a, f'(a) gives the slope of the tangent to the curve.
导数也写作 f'(x) 或 d/dx[f(x)]。在点 x = a 处，f'(a) 给出曲线切线的斜率。
2. Differentiation from First Principles | 从第一性原理微分
Differentiation from first principles uses the gradient of a chord between two close points and then lets the distance between them tend to zero.
从第一性原理微分使用两个相近点之间的弦的斜率，然后让这两点之间的距离趋近于零。
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
This formula is the formal definition of the derivative. For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h².
该公式是导数的正式定义。例如，如果 f(x) = x²，那么 f(x+h) = (x+h)² = x² + 2xh + h²。
Subtracting f(x) and dividing by h gives 2x + h. As h approaches 0, the derivative is 2x.
减去 f(x) 并除以 h 得到 2x + h。当 h 趋近于 0 时，导数为 2x。
3. Standard Derivatives and the Power Rule | 标准导数与幂法则
The power rule is the most common differentiation tool: d/dx(xⁿ) = n xⁿ⁻¹. You can extend this to sums, differences and constant multiples.
幂法则是最常用的微分工具：d/dx(xⁿ) = n xⁿ⁻¹。你可以将其推广到和、差和常数倍。
For example, if y = 4x³ − 2x + 5, then dy/dx = 12x² − 2. The derivative of a constant is always zero.
例如，如果 y = 4x³ − 2x + 5，那么 dy/dx = 12x² − 2。常数的导数始终为零。
| f(x) | f'(x) |
| xⁿ | n xⁿ⁻¹ |
| k | 0 |
| k f(x) | k f'(x) |
| f(x) + g(x) | f'(x) + g'(x) |
4. The Chain Rule | 链式法则
Use the chain rule when differentiating a function of a function. If y = f(u) and u = g(x), then:
当对复合函数求导时使用链式法则。如果 y = f(u) 且 u = g(x)，则：
dy/dx = dy/du × du/dx
For example, let y = (3x² + 5)⁴. Set u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x.
例如，设 y = (3x² + 5)⁴。令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x。
Multiplying gives dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.
相乘得到 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。
5. The Product Rule | 乘积法则
When differentiating the product of two functions u(x) and v(x), use the product rule:
当对两个函数 u(x) 和 v(x) 的乘积求导时，使用乘积法则：
d/dx(uv) = u dv/dx + v du/dx
For example, if y = x² sin x, let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x.
例如，如果 y = x² sin x，令 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x。
Therefore dy/dx = x² cos x + 2x sin x. This is usually the final simplified answer in Edexcel exams.
因此 dy/dx = x² cos x + 2x sin x。在 Edexcel 考试中，这通常作为最终的简化答案。
6. The Quotient Rule | 商法则
To differentiate u(x) divided by v(x), use the quotient rule:
要对 u(x) 除以 v(x) 求导，使用商法则：
d/dx(u/v) = (v du/dx − u dv/dx) / v²
For example, if y = (x² + 1)/(x − 2), let u = x² + 1 and v = x − 2. Then du/dx = 2x and dv/dx = 1.
例如，如果 y = (x² + 1)/(x − 2)，令 u = x² + 1，v = x − 2。则 du/dx = 2x，dv/dx = 1。
So dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (x² − 4x − 1)/(x − 2)².
所以 dy/dx = [(x − 2)(2x) − (x² + 1)(1)] / (x − 2)² = (x² − 4x − 1)/(x − 2)²。
7. Differentiating Exponential, Logarithmic and Trigonometric Functions | 指数、对数与三角函数的微分
You must memorise these standard results for Edexcel A-Level Mathematics:
你必须熟记以下 Edexcel A-Level 数学中的标准导数结果：
d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec² x
When the argument is not simply x, combine the standard result with the chain rule. For example, d/dx(e²ˣ) = 2e²ˣ and d/dx(ln(3x)) = 1/x.
当自变量不只是 x 时，要将标准结果与链式法则结合使用。例如，d/dx(e²ˣ) = 2e²ˣ，d/dx(ln(3x)) = 1/x。
For sin(5x), the derivative is 5 cos(5x). Always check for a multiplying constant from the inner derivative.
对于 sin(5x)，其导数是 5 cos(5x)。始终要检查由内部函数求导产生的乘数。
8. Second Derivatives and Connected Rates of Change | 二阶导数与相关变化率
The second derivative f”(x) or d²y/dx² is the derivative of f'(x). It tells you whether the gradient is increasing or decreasing.
二阶导数 f”(x) 或 d²y/dx² 是 f'(x) 的导数。它告诉你梯度是在增加还是减少。
Connected rates of change use the chain rule with a rate you already know. For example, if A = πr², then dA/dt = dA/dr × dr/dt = 2πr × dr/dt.
相关变化率利用已知的速率与链式法则。例如，如果 A = πr²，那么 dA/dt = dA/dr × dr/dt = 2πr × dr/dt。
This is especially important in modelling questions where a circle’s radius increases at a given rate.
这在建模题中尤为重要，例如圆的半径以给定速率增大的情况。
9. Stationary Points and Optimisation | 驻点与优化
At a stationary point, dy/dx = 0. To determine the nature, use the second derivative test: if f”(x) > 0, it is a local minimum; if f”(x) < 0, it is a local maximum.
在驻点处，dy/dx = 0。要判断其性质，可使用二阶导数检验：如果 f”(x) > 0，则为局部极小值；如果 f”(x) < 0，则为局部极大值。
For example, if f(x) = x³ − 3x + 2, then f'(x) = 3x² − 3. Setting f'(x) = 0 gives x = 1 and x = −1.
例如，如果 f(x) = x³ − 3x + 2，那么 f'(x) = 3x² − 3。令 f'(x) = 0 得到 x = 1 和 x = −1。
Then f”(x) = 6x, so f”(1) = 6 > 0 gives a local minimum, while f”(−1) = −6 < 0 gives a local maximum.
然后 f”(x) = 6x，因此 f”(1) = 6 > 0 给出局部极小值，而 f”(−1) = −6 < 0 给出局部极大值。
10. Common Edexcel Exam Pitfalls | Edexcel 考试常见误区
Many marks are lost by forgetting to apply the chain rule inside standard functions, such as writing d/dx(sin(3x)) = cos(3x) instead of 3 cos(3x).
许多分数因忘记在标准函数内部应用链式法则而丢失，例如把 d/dx(sin(3x)) 写成 cos(3x) 而不是 3 cos(3x)。
Another common mistake is losing the negative sign when differentiating cos x or when applying the quotient rule. Write the quotient rule carefully to avoid sign errors.
另一个常见错误是在对 cos x 求导或应用商法则时丢失负号。仔细书写商法则以避免符号错误。
Always simplify final answers where possible, and check whether the question requires an exact value, a simplified surd, or a unit of measurement.
尽可能简化最终答案，并检查题目是否要求精确值、化简根式或计量单位。
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