📚 Core Differentiation Techniques for Edexcel A-Level Maths | Edexcel A-Level 数学核心微分技巧
Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It underpins applications in optimisation, curve sketching, rates of change and mechanics. This revision guide walks through the core techniques from first principles to parametric differentiation, with clear worked examples, common pitfalls and exam-style tips.
微分是 Edexcel A-Level 数学中考查最频繁的主题之一，也是优化问题、曲线作图、变化率及力学应用的基础。本复习指南从第一性原理到参数微分，系统梳理核心技巧，并配有清晰的示例、常见错误和考试技巧。
1. First Principles | 第一性原理
The derivative of a function f(x) is defined as the limit of the average rate of change as h tends to zero. This definition is the foundation of all differentiation and is tested directly in Edexcel exams.
函数 f(x) 的导数定义为当 h 趋近于零时平均变化率的极限。该定义是所有微分的基础，也是 Edexcel 考试中会直接考查的内容。
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
x², expand (x + h)² = x² + 2xh + h², subtract x², divide by h, then let h → 0 to obtain f'(x) = 2x. This process shows that the gradient of y = x² at any point is 2x.
例如，若 f(x) = x²，展开 (x + h)² = x² + 2xh + h²，减去 x² 后除以 h，再令 h → 0，得到 f'(x) = 2x。这一过程说明 y = x² 的图像在任意一点的斜率为 2x。
too early, before the denominator has been cancelled. Always simplify the fraction fully before taking the limit.
常见错误是在分母尚未约去时就过早代入 h = 0。务必先彻底化简分式，再取极限。
2. Standard Derivatives and the Power Rule | 常见导数与幂法则
nstant. It also applies to negative and fractional powers after rewriting roots and reciprocals.
幂法则指出，若 y = xⁿ，则 dy/dx = n xⁿ⁻¹，其中 n 为任意实常数。该法则同样适用于将根式和倒数改写为负指数和分数指数后的函数。
d/dx (xⁿ) = n xⁿ⁻¹
You must memorise the derivatives of key functions, because mixed questions often combine them with the chain rule, product rule and quotient rule.
必须熟记以下关键函数的导数，因为综合题经常将它们与链式法则、乘法法则和商法则结合使用。
- d/dx (sin x) = cos x | 正弦函数 sin x 的导数为 cos x
- d/dx (cos x) = -sin x | 余弦函数 cos x 的导数为 -sin x
- d/dx (eˣ) = eˣ | 指数函数 eˣ 的导数仍为 eˣ
- d/dx (ln x) = 1/x | 自然对数 ln x 的导数为 1/x
For roots such as √x, rewrite as x1/2 and then apply the power rule. For reciprocals such as 1/x², rewrite as x⁻².
对于根式如 √x，先改写为 x1/2 再应用幂法则。对于倒数如 1/x²，改写为 x⁻² 后求导。
3. Product Rule | 乘法法则
If y = u(x) v(x), then dy/dx = u'(x) v(x) + u(x) v'(x). This rule is needed when two different functions are multiplied together.
若 y = u(x) v(x)，则 dy/dx = u'(x) v(x) + u(x) v'(x)。当两个不同的函数相乘时，需要使用乘法法则。
d/dx (u v) = u' v + u v'
gives dy/dx = 2x sin x + x² cos x. The order of the two terms is not important, but both u'v and uv' must be included.
例如，y = x² sin x 的导数为 dy/dx = 2x sin x + x² cos x。两项的顺序并不重要，但 u'v 和 uv' 都必须包含。
Always state u, u', v and v' clearly before substituting into the formula. This reduces sign errors and makes your working examiner-friendly.
在代入公式前，务必清楚写出 u、u'、v 和 v'。这可以减少符号错误，并使解题过程更清晰。
4. Quotient Rule | 商法则
If y = u(x) / v(x), then dy/dx = [u'(x) v(x) - u(x) v'(x)] / [v(x)]². The minus sign in the numerator is a frequent source of lost marks.
若 y = u(x) / v(x)，则 dy/dx = [u'(x) v(x) - u(x) v'(x)] / [v(x)]²。分子中的负号是常见的失分点。
d/dx (u/v) = (u' v - u v') / v²
For example, y = eˣ / x gives dy/dx = (eˣ · x - eˣ · 1) / x² = eˣ (x - 1) / x². This result can also be obtained by rewriting as eˣ x⁻¹ and using the product rule.
例如，y = eˣ / x 的导数为 dy/dx = (eˣ · x - eˣ · 1) / x² = eˣ (x - 1) / x²。该结果也可以通过将原式改写为 eˣ x⁻¹ 后用乘法法则得到。
For simple quotients such as 1/x or 3/x², it is often quicker to use the power rule after rewriting with negative exponents rather than applying the full quotient rule.
对于 1/x 或 3/x² 等简单分式，通常先改写为负指数再用幂法则，比直接使用完整商法则更快捷。
5. Chain Rule | 链式法则
dx = dy/du × du/dx. The chain rule is essential for composite functions such as sin(3x), e2x and (x² + 1)⁴.
若 y = f(g(x))，设 u = g(x)，则 dy/dx = dy/du × du/dx。链式法则对于 sin(3x)、e2x 和 (x² + 1)⁴ 等复合函数至关重要。
dy/dx = dy/du × du/dx
= (3x² + 5)⁴. Let u = 3x² + 5, then dy/du = 4u³ and du/dx = 6x, so dy/dx = 24x(3x² + 5)³.
例如，y = (3x² + 5)⁴。设 u = 3x² + 5，则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。
A common error is to forget the derivative of the inner function. Always multiply by du/dx after differentiating the outer function.
常见错误是忘记乘以内层函数的导数。在对外层函数求导后，务必乘以 du/dx。
6. Implicit Differentiation | 隐函数微分
with respect to x, treating y as a function of x. Every time you differentiate a y-term, multiply by dy/dx.
当方程同时含有 x 和 y，如 x² + y² = 25 时，对方程两边关于 x 求导，并将 y 视为 x 的函数。每次对含 y 的项求导时，都要乘以 dy/dx。
d/dx (yⁿ) = n yⁿ⁻¹ dy/dx
= 0, so dy/dx = -x/y. This allows you to find gradients without solving for y explicitly.
例如，对 x² + y² = 25 求导，得 2x + 2y dy/dx = 0，因此 dy/dx = -x/y。这样无需显式解出 y 即可求得斜率。
Implicit differentiation is especially useful for circles, ellipses and other curves where y cannot be written as a single function of x.
隐函数微分对于圆、椭圆以及无法将 y 表示为 x 的单一函数等曲线尤为有用。
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