Mastering Differentiation for Edexcel A Level Pure Mathematics | 爱德思 A Level 纯数学微分精讲

📚 Mastering Differentiation for Edexcel A Level Pure Mathematics | 爱德思 A Level 纯数学微分精讲

Differentiation is one of the most heavily examined topics in the Edexcel A Level Pure Mathematics specification. It connects algebraic technique, graphical reasoning and real-world modelling, so examiners regularly combine it with coordinate geometry, trigonometry and optimisation problems.

微分是爱德思 A Level 纯数学考纲中考查频率最高的主题之一。它将代数技巧、图像推理与现实建模联系起来，因此考官经常把它与坐标几何、三角函数和优化问题综合出题。

1. The Gradient of a Curve and First Principles | 曲线的斜率与第一性原理

The derivative measures the instantaneous rate of change of a function. From first principles, it is defined as the limit of the average rate of change as h tends to zero.

导数衡量函数的瞬时变化率。从第一性原理出发，它定义为当 h 趋于 0 时平均变化率的极限。

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

To use this definition, substitute x + h into f, simplify the numerator, cancel h where possible, then let h approach 0. This method is often tested on simple polynomials such as x² or x³.

使用该定义时，将 x + h 代入 f，化简分子，尽可能约去 h，再令 h 趋近于 0。这一方法常针对 x² 或 x³ 等简单多项式进行考查。

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2. Basic Differentiation Rules | 基本求导法则

For a power function y = xⁿ, the derivative is y’ = n xⁿ⁻¹. The constant multiple rule and the sum rule allow term-by-term differentiation.

对于幂函数 y = xⁿ，其导数为 y’ = n xⁿ⁻¹。常数倍法则和和差法则允许逐项求导。

d/dx (xⁿ) = n xⁿ⁻¹

d/dx [a f(x) + b g(x)] = a f ‘(x) + b g ‘(x)

For example, if y = 5x³ − 2x + 7, then dy/dx = 15x² − 2. Constants vanish because their rate of change is zero.

例如，若 y = 5x³ − 2x + 7，则 dy/dx = 15x² − 2。常数项为零，因为它的变化率为零。

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3. Differentiating Standard Functions | 标准函数的导数

Edexcel expects fluency with standard derivatives: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, and d/dx(tan x) = sec² x.

爱德思要求熟练掌握标准函数的导数：d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x，d/dx(sin x) = cos x，d/dx(cos x) = −sin x，以及 d/dx(tan x) = sec² x。

• eˣ: derivative is eˣ；中文：eˣ 的导数为 eˣ
• ln x: derivative is 1/x；中文：ln x 的导数为 1/x
• sin x: derivative is cos x；中文：sin x 的导数为 cos x
• cos x: derivative is −sin x；中文：cos x 的导数为 −sin x
• tan x: derivative is sec² x；中文：tan x 的导数为 sec² x

These results are often combined with the chain, product and quotient rules, so they must be recalled quickly and accurately.

这些结果经常与链式法则、乘积法则和商法则结合使用，因此必须快速准确地回忆。

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4. The Chain Rule | 链式法则

When y is a function of u and u is a function of x, the chain rule states that dy/dx = dy/du × du/dx. It is essential for differentiating composite functions.

当 y 是 u 的函数且 u 是 x 的函数时，链式法则为 dy/dx = dy/du × du/dx。它对复合函数求导至关重要。

dy/dx = dy/du × du/dx

For y = (3x² + 5)⁴, let u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 24x(3x² + 5)³.

例如 y = (3x² + 5)⁴，令 u = 3x² + 5。则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。

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5. The Product Rule | 乘积法则

For y = u v, the product rule is dy/dx = u dv/dx + v du/dx. It should be used when two non-constant factors are multiplied.

对于 y = u v，乘积法则为 dy/dx = u dv/dx + v du/dx。当两个非常数因子相乘时应使用该法则。

dy/dx = u dv/dx + v du/dx

A common mistake is to differentiate the two factors separately and multiply the results. That is incorrect unless one factor is constant.

常见错误是分别对两个因子求导后相乘。除非其中一个因子为常数，否则这种做法是错误的。

For y = x² sin x, set u = x² and v = sin x. Then dy/dx = x² cos x + 2x sin x.

例如 y = x² sin x，设 u = x²，v = sin x。则 dy/dx = x² cos x + 2x sin x。

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6. The Quotient Rule | 商法则

For y = u / v, the quotient rule is dy/dx = (v du/dx − u dv/dx) / v². It is needed when dividing one function by another.

对于 y = u / v，商法则为 dy/dx = (v du/dx − u dv/dx) / v²。当一个函数除以另一个函数时需要使用此法则。

dy/dx = (v du/dx − u dv/dx) / v²

Keep the numerator order as v u’ minus u v’; reversing the order changes the sign and loses marks.

分子顺序必须保持为 v u’ 减去 u v’；颠倒顺序会改变符号并失分。

For y = x / (x + 1), set u = x and v = x + 1. Then dy/dx = ( (x + 1)(1) − x(1) ) / (x + 1)² = 1 / (x + 1)².

例如 y = x / (x + 1)，设 u = x，v = x + 1。则 dy/dx = ( (x + 1)(1) − x(1) ) / (x + 1)² = 1 / (x + 1)²。

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7. The Second Derivative and Concavity | 二阶导数与凹凸性

The second derivative, written d²y/dx², is obtained by differentiating dy/dx. It describes the rate of change of the gradient and indicates concavity.

二阶导数写作 d²y/dx²，由 dy/dx 再次求导得到。它描述斜率的变化率，并反映曲线的凹凸性。

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