📚 Mastering Differentiation for Edexcel A-Level Maths | 精通爱德思A-Level数学微分
Differentiation is the branch of calculus that describes how one quantity changes when another quantity changes. It sits at the heart of A-Level Maths and appears in almost every Pure Mathematics paper, as well as in mechanics and in the modelling questions that link the two. In the Edexcel specification you are expected to differentiate a wide variety of functions, to justify your results, and to apply your skills to problems in geometry, kinematics and optimisation. This article revisits the key techniques - from first principles to parametric equations - so that you are fully prepared for the exam.
微分是A-Level微积分的核心。根据爱德思考试大纲，你需要掌握多种函数的求导，能够说明结果的理由，并能把技能应用于几何、运动学和最优化问题中。本文系统重温微分的关键技巧——从第一原理、幂法则、指数与对数函数、三角函数，到乘积法则、商法则、链式法则、隐函数求导与参数方程求导，再到二阶导数、切线法线、驻点、最优化与相关变化率——确保你为考试做好充分准备。
1. Introduction to Differentiation | 微分简介
f change of the output with respect to the input. If y = f(x), the derivative is written f' r dy/dx. Edexcel uses both notations, so you must be comfortable switching between them: f' e is given as f(x) = ..., while dy/dx is natural when the relationship is given as y = ...
函数 y = f(x) 的导数衡量 y 相对于 x 的瞬时变化率，记作 f'(x) 或 dy/dx。爱德思考试中两种记号都会出现，你必须能够自由切换：当函数以 f(x) = ... 的形式给出时用 f'(x) 更自然，当关系式以 y = ... 的形式给出时用 dy/dx 更自然。
Geometrically, f' part of the specification: once you know the gradient at a point you can write the equation of the tangent, find the equation of the normal, and decide whether a curve is rising, falling or momentarily flat.
几何上，f'(a) 给出曲线 y = f(x) 在 x = a 处切线的斜率。这一个事实打开了考纲中很大一部分内容：只要知道某点的斜率，你就能写出切线方程、求出法线方程，并判断曲线是上升、下降还是暂时平坦。
It is important to separate the average rate of change from the instantaneous rate of change. The average rate over an interval is the gradient of a chord; the instantaneous rate at a point is the gradient of the tangent, obtained by letting the chord shrink to zero length. Although dy/dx is not a fraction, it behaves like one in the chain rule and in connected rates of change, which is why the notation survives.
必须区分平均变化率与瞬时变化率。区间上的平均变化率是弦的斜率；某点的瞬时变化率是切线的斜率，通过让弦的长度趋于零而得到。虽然 dy/dx 并不是分数，但它在链式法则和相关变化率中表现得像分数一样，这正是这套记号能沿用至今的原因。
2. Differentiation from First Principles | 从第一原理求导
The derivative is defined by a limit. This definition is called differentiation from first principles, and Edexcel may ask you to use it directly, especially for a quadratic, a simple cubic or a reciprocal function.
导数由极限定义，这种定义方式称为从第一原理求导。爱德思考试可能直接要求你使用它，尤其是对二次函数、简单三次函数或倒数函数。
f'(x) = limₕ→₀ [ f(x + h) − f(x) ] / h
zero. Never substitute h = 0 in the first line, because the expression is undefined there.
使用定义要按步骤来：先写出差商，通过约掉 h 来化简分子，最后才让 h 趋于 0。千万不要在第一步就代入 h = 0，因为此时表达式没有定义。
= −h / [ x(x + h) ], dividing by h gives −1 / [ x(x + h) ], which tends to −1/x² as h → 0.
对 f(x) = x²，分子为 (x + h)² − x² = 2xh + h²。除以 h 得 2x + h，当 h → 0 时趋于 2x。对 f(x) = 1/x，分子为 1/(x + h) − 1/x = −h / [ x(x + h) ]，除以 h 得 −1 / [ x(x + h) ]，当 h → 0 时趋于 −1/x²。
3. The Power Rule and Polynomials | 幂法则与多项式
The workhorse of A-Level differentiation is the power rule: to differentiate a power of x, multiply by the index and reduce the index by one. This applies to every real index, including negative indices, zero and fractions.
A-Level 微分中最常用的工具是幂法则：对 x 的幂求导，乘以指数，再把指数减一。它对一切实数指数都成立，包括负指数、零指数和分数指数。
d/dx ( xⁿ ) = n xⁿ⁻¹
Differentiation is linear, so you can differentiate a polynomial term by term. Constants differentiate to zero, and a constant multiplier can be taken outside the derivative. Rewrite roots and reciprocals as powers before you differentiate: √x = x1/2 and 1/x² = x⁻².
微分是线性的，因此可以逐项对多项式求导。常数项求导为零，常数因子可以提到导数符号之外。求导前先把根式和倒数改写为幂的形式：√x = x1/2，1/x² = x⁻²。
For example, if y = 3x⁴ − 5x² + 7x − 2 then dy/dx = 12x³ − 10x + 7. If y = √x + 1/x² then dy/dx = 1/(2√x) − 2/x³. Always check the sign of the new index: differentiating a reciprocal makes the negative power one step more negative, never less.
例如，若 y = 3x⁴ − 5x² + 7x − 2，则 dy/dx = 12x³ − 10x + 7。若 y = √x + 1/x²，则 dy/dx = 1/(2√x) − 2/x³。要始终检查新指数的符号：对倒数求导会让负指数变得更负，而不是相反。
4. Exponentials and Logarithms | 指数函数与对数函数
The exponential function eˣ is remarkable because it is its own derivative. When the exponent is linear, the chain rule brings down the coefficient of x, so eᵏˣ differentiates to k eᵏˣ. More generally, aˣ differentiates to aˣ ln a for a positive constant a.
指数函数 eˣ 的特别之处在于它的导数就是它本身。当指数为线性时，链式法则会带下 x 的系数，所以 eᵏˣ 的导数是 k eᵏˣ。更一般地，对正数常数 a，aˣ 的导数是 aˣ ln a。
d/dx ( eˣ ) = eˣ ， d/dx ( ln x ) = 1/x ( x > 0 )
he constant k disappears on differentiation, so ln(5x) also differentiates to 1/x. Combined with the chain rule, ln f(x) differentiates to f'(x) / f(x), a result that appears repeatedly in Edexcel papers.
对对数而言，以 e 为底时导数最简单：ln x 的导数为 1/x。由于 ln(kx) = ln k + ln x，常数 k 在求导时消失，所以 ln(5x) 的导数同样是 1/x。结合链式法则，ln f(x) 的导数为 f'(x) / f(x)，这个结果在爱德思试卷中反复出现。
Worked example: if y = e³ˣ + ln(5x) then dy/dx = 3e³ˣ n a logarithm appears inside a modulus, differentiate the inside and divide by the inside.
例题：若 y = e³ˣ + ln(5x)，则 dy/dx = 3e³ˣ + 1/x。要注意对数的定义域；当题目给出 x > 0 时应加以利用；当对数出现在绝对值里时，先对内部求导再除以内部。
5. Differentiating Trigonometric Functions | 三角函数求导
In Edexcel A-Level Maths, trigonometric calculus is always carried out in radians unless a question explicitly says otherwise. In radians, the derivatives of sine and cosine are clean and have no extra constant factor.
在爱德思 A-Level 数学中，三角函数的微积分一律使用弧度制，除非题目明确说明。在弧度制下，正弦与余弦的导数形式简洁，不会出现额外的常数因子。
d/dx ( sin x ) = cos x ， d/dx ( cos x ) = −sin x ， d/dx ( tan x ) = sec² x
t result follows from writing tan x = sin x / cos x and applying the quotient rule, giving sec² x, which can also be written 1/cos² x.
注意符号：对余弦求导会引入负号，忘记它是最常见的失分点之一。正切的结果可以由 tan x = sin x / cos x 并用商法则推出，得到 sec² x，也可以写成 1/cos² x。
x − cos(x/2) then dy/dx = 3 cos 3x + ½ sin(x/2). Products and quotients of trigonometric functions are handled with the standard rules covered in the next section.
当角度为线性时，链式法则会乘上系数：sin kx 的导数是 k cos kx，cos kx 的导数是 −k sin kx。例如，若 y = sin 3x − cos(x/2)，则 dy/dx = 3 cos 3x + ½ sin(x/2)。三角函数的乘积与商则用下一节的标准法则处理。
6. Product, Quotient and Chain Rules | 乘积法则、商法则与链式法则
Most exam functions are built from simpler ones, so the three combination rules are unavoidable. Identify the structure first: a product, a quotient or a function of a function. Choosing the wrong rule is the fastest way to lose method marks.
考试中的函数大多由简单函数组合而成，因此三条组合法则无法回避。先判断结构：是乘积、商，还是函数的函数。选错法则是丢分最快的方式。
(uv)' = u'v + uv' ， (u/v)' = ( u'v − uv' ) / v² ， dy/dx = (dy/du) × (du/dx)
The product rule is symmetric, so it does not matter which factor you call u; the quotient rule is not, and the numerator must be u'v − uv' in that order. State the formula you are using, list u, u', v and v' explicitly, and substitute carefully - examiners award marks for a correct structure even when the arithmetic slips.
乘积法则是对称的，哪个因子叫 u 都无所谓；商法则不是对称的，分子必须是 u'v − uv' 这一顺序。写出所用的公式，明确列出 u、u'、v 和 v'，再仔细代入——即使算术出错，写出正确结构也能拿分。
)⁴. In the last case, notice how much faster the chain rule is than expanding the bracket.
值得记住的例子：若 y = x² sin x，则 dy/dx = 2x sin x + x² cos x；若 y = ln x / x，则 dy/dx = (1 − ln x) / x²；若 y = (3x + 1)⁵，则 dy/dx = 15(3x + 1)⁴。最后一例说明链式法则比展开括号快得多。
7. Implicit Differentiation | 隐函数求导
When y is not given explicitly as a function of x, you differentiate both sides of the equation with respect to x. Every time you differentiate a term containing y, the chain rule produces a factor of dy/dx, which you then collect and factorise.
当 y 没有以 x 的显函数形式给出时，就对等式两边同时关于 x 求导。每当对含 y 的项求导时，链式法则会产生一个 dy/dx 因子，之后再把它收集并提取公因式。
x² + y² = 25 ⟹ 2x + 2y (dy/dx) = 0 ⟹ dy/dx = −x/y
al gradient you substitute both coordinates of the point. The gradient is undefined where y = 0, which corresponds to the vertical tangents at the ends of the horizontal diameter.
圆的例子最经典。注意导数是同时用 x 和 y 表示的，这完全可以接受；要得到具体的斜率，就把该点的两个坐标都代入。当 y = 0 时斜率无定义，对应水平直径两端的竖直切线。
Always differentiate the whole term, not just the visible part, and never forget the dy/dx on a y-term.
对更复杂的表达式，如 x²y + y³ = 10，第一项需要乘积法则：2xy + x² (dy/dx) +
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