📚 Mastering Differentiation for Edexcel A-Level Maths | Edexcel A-Level 数学微分核心突破
Differentiation is a central topic in Edexcel A-Level Mathematics, appearing in Pure 1, Pure 2 and Pure 3. It is the tool we use to understand how a function changes at an instant, and it underpins many applications such as curve sketching, optimisation and kinematics. This revision note walks through the key rules, standard derivatives and exam applications, with paired explanations in English and Chinese to help you revise efficiently.
微分是 Edexcel A-Level 数学的核心主题，出现在 Pure 1、Pure 2 和 Pure 3 中。我们用微分来理解函数在某一瞬间的变化方式，它是曲线作图、最优化和运动学等应用的基础。本篇复习笔记梳理关键法则、标准导数与考试应用，并采用中英对照讲解，帮助你高效复习。
1. What Differentiation Measures | 微分度量什么
Differentiation answers one question: how fast is y changing when x changes? = f(x) at that point. It measures the instantaneous rate of change of y with respect to x.
微分回答一个核心问题：当 x 变化时，y 变化得有多快？函数 f(x) 在点 x 处的导数，就是曲线 y = f(x) 在该点处切线的斜率，它度量 y 对 x 的瞬时变化率。
If the derivative is positive, the function is increasing at that point; if negative, it is decreasing; if it is zero, the tangent is horizontal and the point is stationary, which may be a maximum, a minimum or a point of inflection.
如果导数为正，函数在该点递增；如果为负，则递减；如果为零，切线水平，该点是驻点，可能是极大值点、极小值点或拐点。
Compare this with the average rate of change, which is the gradient of a chord joining two points. Differentiation takes the limit as the two points merge together, so a chord becomes a tangent. This is exactly why the gradient of a curve is only defined "at a point".
请与平均变化率对比：平均变化率是连接两点的弦的斜率。微分让两点无限接近并取极限，于是弦变成了切线。这正是"曲线的斜率"只能定义在某一点的原因。
2. First Principles and the Limit Definition | 第一原理与极限定义
The formal definition of the derivative comes from first principles. The gradient of the chord between (x, f(x)) and (x + h, f(x + h)) is [f(x + h) - f(x)] / h. Letting h shrink towards zero turns the chord into the tangent at x.
导数的正式定义来自第一原理。连接 (x, f(x)) 与 (x + h, f(x + h)) 两点的弦的斜率为 [f(x + h) - f(x)] / h。令 h 趋向于零，这条弦就变成 x 处的切线。
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
hen let h → 0. For f(x) = x², the fraction becomes (x² + 2xh + h² − x²) / h = 2x + h, so f'(x) = 2x.
所有第一原理题目都遵循同样的四步：展开 f(x + h)；减去 f(x)；除以 h 并化简；令 h → 0。对 f(x) = x²，该分式化为 (x² + 2xh + h² − x²) / h = 2x + h，因此 f'(x) = 2x。
In the exam, a first-principles question is usually worth 3 or 4 marks, and you must show every algebraic step with the limit notation still visible on the final line. Simply writing the answer by quoting the power rule scores nothing.
考试中第一原理题通常值 3 到 4 分，你必须写出每一步代数过程，并在最后一行保留极限记号。如果直接引用幂函数法则写出答案，是拿不到分的。
3. Notation: dy/dx, f'(x) and y' | 记法：dy/dx、f'(x) 与 y'
Edexcel uses three notations for the same idea. dy/dx is Leibniz notation and emphasises the variables involved; f'(x) is function notation and emphasises the function being differentiated; y' is a shorthand that appears in mechanics and in some textbook solutions. You should be comfortable switching between all three mid-question.
Edexcel 用三种记法表达同一个概念。dy/dx 是莱布尼茨记法，强调所涉及的变量；f'(x) 是函数记法，强调被求导的函数；y' 是简写形式，常见于力学与部分教材解答。你应当能在解题过程中自如切换这三者。
For second derivatives, write d²y/dx², f''(x) or y''. The second derivative tells you how the gradient itself is changing, which is why it is used to classify stationary points.
二阶导数写作 d²y/dx²、f''(x) 或 y''。二阶导数描述斜率本身如何变化，因此常用来判定驻点的类型。
nto the derivative, not into the original function. For example, if f(x) = x³ − 4x, then f'(x) = 3x² − 4, so f'(2) = 8 and the gradient at x = 2 is 8.
当题目要求某一点的斜率时，要代入导数而不是原函数。例如 f(x) = x³ − 4x，则 f'(x) = 3x² − 4，所以 f'(2) = 8，即 x = 2 处的斜率为 8。
4. Differentiating Powers of x | x 的幂函数求导
ere n can be any rational number, positive or negative, integer or fraction. Multiply by the old power, then reduce the power by one.
Pure 1 与 Pure 2 的主力法则是：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹，其中 n 可以是任意有理数，正负皆可，整数或分数皆可。先乘以原来的指数，再把指数减一。
Roots and reciprocals must be rewritten as powers of x before you differentiate. This rewriting step is where most marks are lost, so do it carefully and in writing.
根式与倒数必须先改写为 x 的幂次再求导。这一改写步骤是失分最多的地方，请务必写下来、写清楚。
| Original form | Rewrite as a power | Derivative |
| √x | x^½ | ½x−½ = 1 / (2√x) |
| 1 / x | x⁻¹ | −x⁻² = −1 / x² |
| 1 / x² | x⁻² | −2x⁻³ = −2 / x³ |
| ∛x | x1/3 | (1/3)x−2/3 |
zero gradient. A constant multiplier simply stays in front: if y = 7x⁴ then dy/dx = 28x³.
任何常数项求导结果为零，因为水平线的斜率为零。常数系数则原样保留：若 y = 7x⁴，则 dy/dx = 28x³。
5. Sums, Differences and Constant Multiples | 和、差与常数倍
Differentiation is linear, which means you can differentiate a polynomial term by term. There is no need for a product rule when two expressions are added or subtracted: split them up and handle each separately.
微分是线性的，这意味着多项式可以逐项求导。当两个表达式相加或相减时无需使用乘积法则：拆开来分别处理即可。
he constant −2 disappears. Check that every power has dropped by one and that every coefficient has been multiplied.
对 y = 4x³ − 5x² + 7x − 2，逐项求导得 dy/dx = 12x² − 10x + 7，常数 −2 消失。请检查每个指数是否都减了一，每个系数是否都乘过了。
nd then dy/dx = 9x² − 10x. Expanding here is faster and safer than using the product rule.
若括号与多项式相乘，只要展开方便就先展开。例如 y = x²(3x − 5) 可写成 y = 3x³ − 5x²，于是 dy/dx = 9x² − 10x。此时展开比用乘积法则更快也更稳妥。
6. Standard Derivatives Beyond Powers | 幂函数之外的标准导数
rivatives that must be memorised. The exponential function is its own derivative: if y = eˣ then dy/dx = eˣ, and more generally y = ekx gives dy/dx = k ekx.
Pure 2 与 Pure 3 要求背下一小组标准导数。指数函数的导数就是它自己：若 y = eˣ，则 dy/dx = eˣ；更一般地，y = ekx 给出 dy/dx = k ekx。
The natural logarithm gives y = ln x ⟹ often combined with the chain rule to differentiate ln(ax + b), which gives a / (ax + b).
自然对数给出 y = ln x ⟹ dy/dx = 1 / x，定义域为 x > 0。该结果常与链式法则结合，用于求 ln(ax + b) 的导数，结果为 a / (ax + b)。
For trigonometry, y = sin x ⟹ dy/dx = cos x and y = cos x ⟹ and is one of the most frequently dropped marks in the whole specification. Also y = tan x ⟹ dy/dx = sec²x.
三角函数方面，y = sin x ⟹ dy/dx = cos x，y = cos x ⟹ dy/dx = −sin x。这个负号至关重要，也是整个考纲中最常被漏掉的分之一。此外 y = tan x ⟹ dy/dx = sec²x。
7. Chain, Product and Quotient Rules | 链式、乘积与商法则
composite functions. In the form dy/dx = dy/du × du/dx, or equivalently d/dx[f(g(x))] = f'(g(x)) × g' : differentiate the outer function, then multiply by the derivative of the inner function.
链式法则处理复合函数。写作 dy/dx = dy/du × du/dx，或等价地 d/dx[f(g(x))] = f'(g(x)) × g'(x)，其含义是：先对外层函数求导，再乘以内层函数的导数。
/dx = 5(3x + 1)⁴ × 3 = 15(3x + 1)⁴. Likewise y = sin(4x) gives dy/dx = 4cos(4x), and y = e2x + 1 gives dy/dx = 2e2x + 1.
例如 y = (3x + 1)⁵ 给出 dy/dx = 5(3x + 1)⁴ × 3 = 15(3x + 1)⁴。同样地，y = sin(4x) 给出 dy/dx = 4cos(4x)，y = e2x + 1 给出 dy/dx = 2e2x + 1。
ies to a product of two functions: if y = uv then dy/dx = u(dv/dx) + v(du/dx). For y = x²eˣ, take u = x² and v = eˣ to obtain dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2).
乘积法则适用于两个函数的乘积：若 y = uv，则 dy/dx = u(dv/dx) + v(du/dx)。对 y = x²eˣ，取 u = x²、v = eˣ，得 dy/dx = x²eˣ + 2xeˣ = xeˣ(x + 2)。
le, y = (2x + 1) / (x − 3) gives dy/dx = [2(x − 3) − (2x + 1)] / (x − 3)² = −7 / (x − 3)².
商法则适用于分式：若 y = u / v，则 dy/dx = [v(du/dx) − u(dv/dx)] / v²。例如 y = (2x + 1) / (x − 3) 给出 dy/dx = [2(x − 3) − (2x + 1)] / (x − 3)² = −7 / (x − 3)²。
8. Tangents, Normals and Gradients at a Point | 切线、法线与点处斜率
The gradient of the curve y = f(x) at x = a is f' m and the point (a, f(a)), the equation of the tangent is found from y − f(a) = m(x − a).
曲线 y = f(x) 在 x = a 处的斜率是 f'(a)。已知斜率 m 和点 (a, f(a)) 后，切线方程由 y − f(a) = m(x − a) 得到。
adient is −1 / m. For y = x³ − 3x at x = 2, the point is (2, 2), the tangent gradient is f'(2) = 9, the tangent is y = 9x − 16 and the normal is y = (−1/9)x + 20/9.
法线是在同一点与切线垂直的直线，因此其斜率为 −1 / m。对 y = x³ − 3x 在 x = 2 处，点为 (2, 2)，切线斜率为 f'(2) = 9，切线为 y = 9x − 16，法线为 y = (−1/9)x + 20/9。
Questions sometimes say the tangent is parallel to a given line. In that case set f'(x) equal to the gradient of the line and solve for x. If the tangent is perpendicular to the line, set f'(x) equal to the negative reciprocal of its gradient.
题目有时会说切线与某条已知直线平行。此时令 f'(x) 等于该直线的斜率并解出 x。若切线与该直线垂直，则令 f'(x) 等于其斜率的负倒数。
