Differentiation in Edexcel A-Level Pure Maths | 爱德思 A-Level 纯数学中的微分

📚 Differentiation in Edexcel A-Level Pure Maths | 爱德思 A-Level 纯数学中的微分

Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It underpins applications such as curve sketching, optimisation, motion, and connected rates of change. Mastering the core rules and recognising when to apply them is essential for both AS and A-Level Pure Papers.

微分是爱德思 A-Level 纯数学中考查频率最高的主题之一。它是曲线作图、最优化、运动学以及相关变化率等应用的基础。掌握核心法则并能在不同情境中准确判断使用哪种方法是 AS 与 A-Level 纯数学试卷取得高分的关键。

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1. Differentiation from First Principles | 第一原理微分

The derivative of f(x) is defined as f'(x) = limₕ→₀ [f(x + h) − f(x)] / h. Edexcel often asks candidates to prove the derivative of simple functions such as x² or x³ from first principles. You must show each step: substitute into the formula, expand brackets, simplify, cancel h, and let h tend to 0.

导数定义为 f'(x) = limₕ→₀ [f(x + h) − f(x)] / h。爱德思考试常要求考生从第一原理证明 x² 或 x³ 等简单函数的导数。你需要写出每一步：代入公式、展开括号、化简、约去 h，再令 h 趋近于 0。

For example, to differentiate f(x) = x² from first principles:

例如，从第一原理出发求 f(x) = x² 的导数：

f'(x) = limₕ→₀ [(x + h)² − x²] / h = limₕ→₀ (2xh + h²) / h = limₕ→₀ (2x + h) = 2x

The final step is valid because h tends to 0, leaving 2x as the derivative. Always write the limit symbol until the moment h is removed.

最后一步之所以成立，是因为 h 趋近于 0，留下 2x 作为导数。在 h 被消去之前，要一直保留极限符号。

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2. Standard Derivatives and Basic Rules | 标准导数与基本法则

Before tackling harder problems, you must be able to recall standard derivatives instantly. The table below summarises the results most commonly required in Edexcel papers.

在处理更复杂的问题之前，你必须能够快速回忆起标准导数公式。下表总结了爱德思考试中最常要求的结果。

f(x)	f'(x)
xⁿ	n xⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec² x
c (constant)	0

Differentiation is linear: if f(x) = a g(x) + b h(x), then f'(x) = a g'(x) + b h'(x). This means constants can be taken outside the derivative, and terms can be differentiated separately.

微分运算具有线性性质：如果 f(x) = a g(x) + b h(x)，那么 f'(x) = a g'(x) + b h'(x)。这意味着常数可以提到导数符号外，各项可以分别求导。

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3. Product Rule | 积法则

If y = u(x)v(x), then dy/dx = u’v + uv’. It is used when both factors are functions of x, such as y = x² eˣ or y = x sin x. Always identify u and v clearly before differentiating.

如果 y = u(x)v(x)，则 dy/dx = u’v + uv’。当两个因子都是 x 的函数时，如 y = x² eˣ 或 y = x sin x，就需要使用积法则。微分前要明确写出 u 和 v。

Example: differentiate y = x² eˣ.

示例：求 y = x² eˣ 的导数。

u = x², v = eˣ, u’ = 2x, v’ = eˣ so dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x)

In an exam, always quote the formula first and then substitute the four pieces u’, v, u, and v’. Simplification is normally expected unless the question says otherwise.

在考试中，要先写出公式，再代入 u’、v、u、v’ 四个部分。除非题目另有说明，通常需要化简结果。

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4. Quotient Rule | 商法则

If y = u(x)/v(x), then dy/dx = (u’v − uv’) / v². Edexcel often includes rational functions such as y = x/(x² + 1) or trigonometric quotients. Watch the subtraction order: the derivative of the numerator comes first.

如果 y = u(x)/v(x)，则 dy/dx = (u’v − uv’) / v²。爱德思考试常包含有理函数，如 y = x/(x² + 1) 或三角函数的商。注意减法顺序：分子的导数在前。

Example: differentiate y = x / (x² + 1).

示例：求 y = x / (x² + 1) 的导数。

u = x, v = x² + 1, u’ = 1, v’ = 2x, so dy/dx = [(1)(x² + 1) − (x)(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)²

The denominator is always squared, not differentiated. A common mistake is to write v’ in the denominator or to reverse the order in the numerator.

分母始终需要平方，而不是对其求导。常见错误是在分母中写 v’，或者把分子中的减法顺序颠倒。

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5. Chain Rule | 链式法则

The chain rule is used for composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In alternative notation, dy/dx = dy/du × du/dx. This is essential for powers of brackets, such as y = (3x² + 5)⁴, and for functions of linear forms, such as y = sin(2x + 1).

链式法则用于复合函数：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。用另一种记号表示为 dy/dx = dy/du × du/dx。这对带括号的幂函数（如 y = (3x² + 5)⁴）和线性形式的函数（如 y = sin(2x + 1)）都至关重要。

Example: differentiate y = (3x² + 5)⁴.

示例：求 y = (3x² + 5)⁴ 的导数。

Let u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³

The key idea is to differentiate the outer function first, leave the inner bracket unchanged, and then multiply by the derivative of the inner bracket.

关键思想是先对外层函数求导，内层括号暂时保持不变，然后再乘以内层括号的导数。

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6. Differentiating Exponentials, Logarithms and Trigonometric Functions | 指数、对数与三角函数的微分

For eᵏˣ, the derivative is k eᵏˣ. For ln(g(x)), the derivative is g'(x)/g(x); in particular, d/dx ln(kx) = 1/x. For sin(kx) and cos(kx), use the chain rule to obtain k cos(kx) and −k sin(kx). These results appear frequently in applied contexts.

对于 eᵏˣ，导数为 k eᵏˣ。对于 ln(g(x))，导数为 g'(x)/g(x)；特别地，d/dx ln(kx) = 1/x。对于 sin(kx) 和 cos(kx)，应用链式法则得到 k cos(kx) 和 −k sin(kx)。这些结果在应用题中频繁出现。

Example: differentiate y = ln(2x + 3).

示例：求 y = ln(2x + 3) 的导数。

dy/dx = 2 / (2x + 3)

Do not forget the derivative of the inner linear expression. Similarly, d/dx e³ˣ = 3e³ˣ and d/dx cos(4x) = −4 sin(4x).

不要忘记内层线性表达式的导数。类似地，d/dx e³ˣ = 3e³ˣ，d/dx cos(4x) = −4 sin(4x)。

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7. Implicit Differentiation | 隐函数微分

When y is not given explicitly as a function of x, differentiate both sides with respect to x and multiply every derivative of y by dy/dx. For example, differentiating x² + y² = 25 gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. Edexcel often tests implicit differentiation combined with tangents and normals.

当 y 没有显式表示为 x 的函数时，对等式两边关于 x 求导，并将每一个 y 的导数乘以 dy/dx。例如，对 x² + y² = 25 求导得到 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。爱德思常将隐函数微分与切线和法线结合考查。

Example: find dy/dx if x³ + 2xy + y² = 10.

示例：若 x³ + 2xy + y² = 10，求 dy/dx。

3x² + 2(y + x dy/dx) + 2y dy/dx = 0 ⇒ 3x² + 2y + (2x + 2y) dy/dx = 0

dy/dx = −(3x² + 2y) / (2x + 2y)

Remember to use the product rule for terms like 2xy where both x and y appear. Collect all dy/dx terms on one side before factorising.

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