📚 Edexcel A Level Maths: Differentiation Core Skills | 爱德思A-Level数学：微分核心技能
Differentiation is one of the two central ideas in calculus, alongside integration. At A-level, Edexcel students use differentiation to find gradients of curves, rates of change, equations of tangents and normals, stationary points, and acceleration in kinematics.
微分与积分是微积分中的两大核心思想。在 A-level 阶段，爱德思考生利用微分来求曲线斜率、变化率、切线与法线方程、驻点以及运动学中的加速度。
This revision guide covers the key methods and exam techniques required for the Edexcel Pure Mathematics specification. The focus is on clean notation, first-principles work, and reliable classification of stationary points.
本复习指南涵盖爱德思纯数学考试所需的关键方法与考试技巧，重点在于清晰的记号、第一原理运算以及驻点的可靠分类。
1. Gradient of a Curve and Limits | 曲线的梯度与极限
A straight-line graph has a constant gradient, but most curves do not. The gradient of a curve at a point is the gradient of the tangent to the curve at that point. To find this gradient precisely, we use a limiting process.
直线图像的梯度是常数，但大多数曲线的梯度不是。曲线上某一点的梯度就是该点切线的梯度。为了精确求出这个梯度，我们需要使用极限过程。
Consider a chord joining two nearby points on the curve y = f(x). As the second point moves closer to the first, the chord becomes a better approximation to the tangent. The limiting value of the chord’s gradient is the derivative.
考虑连接曲线 y = f(x) 上两个邻近点的弦。当第二个点向第一个点无限靠近时，弦越来越接近切线。弦的斜率的极限值就是导数。
slope of chord = [f(x + h) − f(x)] / h
2. Differentiation from First Principles | 从第一原理求导
Differentiation from first principles uses the limit definition. If f(x) is differentiable at x, then f'(x) = limₕ→₀ [f(x + h) − f(x)] / h, provided this limit exists.
从第一原理求导使用极限定义。如果 f(x) 在 x 处可导，那么 f'(x) = limₕ→₀ [f(x + h) − f(x)] / h，前提是该极限存在。
To differentiate x² from first principles, expand (x + h)² = x² + 2xh + h², subtract x², divide by h, and let h tend to 0. This gives 2x.
从第一原理对 x² 求导时，展开 (x + h)² = x² + 2xh + h²，减去 x²，再除以 h，并令 h 趋向于 0。这样就得到 2x。
f'(x) = limₕ→₀ (2xh + h²) / h = 2x
3. Derivative Notation | 导数记号
There are several common notations for the derivative. If y = f(x), the derivative can be written as f'(x), dy/dx, or d/dx[f(x)]. The notation dy/dx is read as “dy by dx” and emphasises the rate of change of y with respect to x.
导数有几种常见的记号。如果 y = f(x)，导数可以写成 f'(x)、dy/dx 或 d/dx[f(x)]。记号 dy/dx 读作 “dy by dx”，强调 y 对 x 的变化率。
In mechanics, when differentiating with respect to time t, Newton’s dot notation is sometimes used, but dx/dt and d²x/dt² are standard in Edexcel questions. In pure mathematics, f'(x) and dy/dx are the most common forms.
在力学中，对时间 t 求导时有时会使用牛顿的点记号，但爱德思试题中标准写法是 dx/dt 和 d²x/dt²。在纯数学中，f'(x) 和 dy/dx 是最常见的形式。
4. The Power Rule | 幂法则
The power rule states that if f(x) = xⁿ for any real number n, then f'(x) = nxⁿ⁻¹. This is the most important rule in elementary differentiation and is valid for positive integers, negative integers, and fractional powers.
幂法则指出，如果 f(x) = xⁿ（n 为任意实数），那么 f'(x) = nxⁿ⁻¹。这是初等微分中最重要的法则，适用于正整数、负整数和分数次幂。
For example, d/dx(x⁵) = 5x⁴, d/dx(x⁻²) = −2x⁻³, and d/dx(√x) = 1 / (2√x). Always rewrite roots as powers if this helps you apply the rule consistently.
例如，d/dx(x⁵) = 5x⁴，d/dx(x⁻²) = −2x⁻³，以及 d/dx(√x) = 1 / (2√x)。如果这有助于你一致地应用法则，请始终将根式改写为幂的形式。
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
5. Constant Multiple, Sum and Difference Rules | 常数倍、和与差法则
If f(x) and g(x) are differentiable, then d/dx[af(x) + bg(x)] = a f'(x) + b g'(x), where a and b are constants. This linearity rule allows us to differentiate term by term.
如果 f(x) 和 g(x) 可导，那么 d/dx[af(x) + bg(x)] = a f'(x) + b g'(x)，其中 a 和 b 为常数。这条线性法则允许我们逐项求导。
Constant multiple: d/dx[5x³] = 15x². Sum rule: d/dx[x³ + x²] = 3x² + 2x. The derivative of a constant alone is zero because a constant does not change.
常数倍法则：d/dx[5x³] = 15x²。和法则：d/dx[x³ + x²] = 3x² + 2x。常数单独的导数为零，因为常数不变化。
6. Differentiating Polynomials | 多项式求导
To differentiate a polynomial, apply the power rule to each term separately. Example: if y = 4x³ − 3x² + 2x − 7, then dy/dx = 12x² − 6x + 2.
对多项式求导时，对每一项分别应用幂法则。例如，如果 y = 4x³ − 3x² + 2x − 7，那么 dy/dx = 12x² − 6x + 2。
Always expand brackets or simplify before differentiating where possible. For products such as (x + 2)(x − 3), expand to x² − x − 6 first, then differentiate to 2x − 1.
在可能的情况下，求导前先展开括号或化简。例如对于乘积 (x + 2)(x − 3)，先展开为 x² − x − 6，再求导得到 2x − 1。
y = 4x³ − 3x² + 2x − 7 ⇢ dy/dx = 12x² − 6x + 2
7. Tangents and Normals | 切线与法线
Once the derivative is known, we can find the equation of a tangent or normal at a given point. If the curve y = f(x) has derivative m = f'(a) at x = a, then the tangent at (a, f(a)) has gradient m.
一旦求出导数，我们就能求曲线在某点的切线或法线方程。如果曲线 y = f(x) 在 x = a 处的导数为 m = f'(a)，那么点 (a, f(a)) 处的切线斜率就是 m。
The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0. Use the point-gradient form y − y₁ = m(x − x₁) for both the tangent and the normal.
法线垂直于切线，因此其斜率为 −1/m（前提 m ≠ 0）。切线和法线都使用点斜式 y − y₁ = m(x − x₁）。
Tangent: y − y₁ = m(x − x₁); Normal: y − y₁ = (−1/m)(x − x₁)
8. The Second Derivative | 二阶导数
The second derivative is the derivative of the derivative. It is written as f”(x) or d²y/dx². It measures the rate of change of the gradient, often interpreted as acceleration in kinematics.
二阶导数是导数的导数。它写作 f”(x
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