Mastering Differentiation for Edexcel A-Level Pure Mathematics | 掌握 Edexcel A-Level 纯数学微分

📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | 掌握 Edexcel A-Level 纯数学微分

Differentiation is one of the most important topics in the Edexcel A-Level Pure Mathematics specification. It underpins many applications across mechanics, statistics and real-world modelling. This article provides a structured revision guide to the key techniques of differentiation, common exam-style applications and the mistakes you should avoid.

微分是 Edexcel A-Level 纯数学考试大纲中最重要的主题之一。它是力学、统计以及现实建模中许多应用的基础。本文为微分的关键技巧、常见考试应用以及需要避免的错误提供结构化复习指南。

1. Differentiation as a Gradient Function | 微分作为梯度函数

The derivative of a function f(x) is written as f'(x) or dy/dx when y = f(x). It represents the instantaneous rate of change of y with respect to x. Geometrically, f'(a) is the gradient of the tangent to the curve y = f(x) at x = a.

函数 f(x) 的导数写作 f'(x)；当 y = f(x) 时也可写作 dy/dx。它表示 y 关于 x 的瞬时变化率。从几何上看，f'(a) 是曲线 y = f(x) 在 x = a 处切线的斜率。

A positive derivative means the function is increasing at that point; a negative derivative means it is decreasing. This sign interpretation is essential for analysing graphs and optimisation.

导数为正表示函数在该点递增；导数为负表示函数在该点递减。这种符号解释对于分析图形和最优化问题至关重要。

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

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2. Differentiation from First Principles | 从第一原理出发求导

The formal definition of the derivative uses a limit. For a function f(x), the derivative is defined as f'(x) = lim (h → 0) [f(x+h) − f(x)] / h. This expression measures the gradient of a chord as the two points on the curve become infinitely close.

导数的正式定义用到了极限。对于函数 f(x)，导数定义为 f'(x) = lim (h → 0) [f(x+h) − f(x)] / h。该表达式衡量曲线上两点无限接近时割线的斜率。

Edexcel often asks candidates to differentiate simple polynomials such as x² or x³ from first principles. You should expand brackets, simplify the numerator, cancel h and then let h tend to zero.

Edexcel 经常要求考生从第一原理出发求 x² 或 x³ 等简单多项式的导数。你应当展开括号、化简分子、约去 h，然后令 h 趋于零。

For f(x) = x²: f'(x) = lim (h → 0) [(x+h)² − x²] / h = 2x

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3. The Power Rule | 幂法则

For any real constant n, if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This rule is the most frequently used differentiation technique and applies to positive, negative and fractional powers.

对于任意实数常量 n，如果 f(x) = xⁿ，那么 f'(x) = n xⁿ⁻¹。该法则是使用频率最高的微分技巧，适用于正整数、负整数以及分数次幂。

f(x) = x⁴	f'(x) = 4x³
f(x) = x⁻²	f'(x) = −2x⁻³
f(x) = √x = x¹⁄²	f'(x) = (1/2) x⁻¹⁄²

Always rewrite roots and reciprocals as powers of x before differentiating. This reduces errors and makes the application of the power rule straightforward.

在求导之前，始终将根式和倒数改写为 x 的幂形式。这能减少错误，并让幂法则的应用更加直接。

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4. Constant Multiple and Sum-Difference Rules | 常数倍法则与和差法则

If f(x) = c g(x), where c is a constant, then f'(x) = c g'(x). Also, the derivative of a sum or difference is the sum or difference of the derivatives: d/dx[u(x) ± v(x)] = u'(x) ± v'(x).

如果 f(x) = c g(x)，其中 c 是常数，那么 f'(x) = c g'(x)。此外，和或差的导数等于导数的和或差：d/dx[u(x) ± v(x)] = u'(x) ± v'(x)。

These rules allow you to differentiate polynomials term by term. For example, if y = 3x⁴ − 5x² + 2x − 7, then dy/dx = 12x³ − 10x + 2.

这些法则允许你逐项对多项式求导。例如，如果 y = 3x⁴ − 5x² + 2x − 7，那么 dy/dx = 12x³ − 10x + 2。

d/dx[3x⁴ − 5x² + 2x − 7] = 12x³ − 10x + 2

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5. Higher-Order Derivatives | 高阶导数

The second derivative, written as f”(x) or d²y/dx², is obtained by differentiating f'(x). It measures the rate of change of the gradient and helps determine the nature of stationary points.

二阶导数写作 f”(x) 或 d²y/dx²，是由 f'(x) 再次求导得到的。它衡量斜率的变化率，并帮助判断驻点的性质。

If f”(a) > 0 at a stationary point, the point is a local minimum. If f”(a) < 0, it is a local maximum. When f''(a) = 0, further investigation is required.

如果在驻点处 f”(a) > 0，该点为局部极小值点；如果 f”(a) < 0，该点为局部极大值点。若 f''(a) = 0，则需要进一步判断。

For example, if y = x³ − 3x, then y’ = 3x² − 3 and y” = 6x. At x = 1, y” = 6 > 0, so the point is a local minimum.

例如，如果 y = x³ − 3x，那么 y’ = 3x² − 3，y” = 6x。在 x = 1 处，y” = 6 > 0，因此该点为局部极小值点。

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6. The Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. In simpler form, if y = [g(x)]ⁿ, then dy/dx = n[g(x)]ⁿ⁻¹ g'(x).

链式法则用于复合函数求导。如果 y = f(u) 且 u = g(x)，那么 dy/dx = dy/du × du/dx。简化为：若 y = [g(x)]ⁿ，则 dy/dx = n[g(x)]ⁿ⁻¹ g'(x)。

For example, if y = (2x³ + 5)⁴, set u = 2x³ + 5. Then dy/du = 4u³ and du/dx = 6x², so dy/dx = 24x²(2x³ + 5)³.

例如，如果 y = (2x³ + 5)⁴，令 u = 2x³ + 5。则 dy/du = 4u³，du/dx = 6x²，因此 dy/dx = 24x²(2x³ + 5)³。

dy/dx = dy/du × du/dx

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7. Product and Quotient Rules | 乘积法则与商法则

For two differentiable functions u(x) and v(x), the product rule states that d/dx(uv) = u’v + uv’. The quotient rule states that d/dx(u/v) = (u’v − uv’) / v², provided v ≠ 0.

对于两个可导函数 u(x) 和 v(x)，乘积法则为 d/dx(uv) = u’v + uv’。商法则为 d/dx(u/v) = (u’v − uv’) / v²，其中 v ≠ 0。

A simple product example:

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