Differentiation Techniques for Edexcel A-Level Mathematics | 爱德思A-Level数学微分技巧精讲

📚 Differentiation Techniques for Edexcel A-Level Mathematics | 爱德思A-Level数学微分技巧精讲

Differentiation is one of the central pillars of Edexcel A-Level Pure Mathematics. It is used to find gradients of curves, equations of tangents and normals, rates of change, optimisation problems, and to interpret real-world motion. This article reviews the main differentiation techniques you need to master for the exam, with key formulas, examples and common pitfalls.

微分是爱德思A-Level纯数学的核心支柱之一。它用于求解曲线的梯度、切线与法线方程、变化率、优化问题以及解释现实中的运动。本文回顾考试必须掌握的主要微分技巧，给出关键公式、例子和常见易错点。

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1. The Gradient Function and First Principles | 导函数与第一原理

ifference quotient. You may be asked to prove this for simple functions such as f(x) = x².

函数 f(x) 的导数给出了曲线在任意点处切线的斜率。它从第一原理出发定义为差商的极限。考试可能要求你对简单函数（如 f(x) = x²）进行证明。

f'(x) = lim (h → 0) [f(x+h) − f(x)] / h

lifies to 2x + h, and as h tends to 0, the limit is 2x. This confirms that d/dx (x²) = 2x.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。差商化简为 2x + h，当 h 趋于 0 时极限为 2x。这验证了 d/dx (x²) = 2x。

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2. Standard Derivatives and the Power Rule | 标准导数与幂法则

The power rule states that if f(x) = xⁿ, then f' and constant multiples, so you can differentiate polynomials term by term with confidence.

幂法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。该法则可应用于和、差和常数倍，因此你可以逐项可靠地对多项式进行微分。

The table below summarises the standard derivatives that are used throughout the Edexcel A-Level course.

下表总结了整个爱德思A-Level课程中使用的标准导数。

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

Remember that a constant alone differentiates to zero, and expressions such as 1/x and √x should be rewritten as x⁻¹ and x^1/2 before applying the power rule.

请记住，单独的常数微分后为零，而诸如 1/x 和 √x 的表达式应改写为 x⁻¹ 和 x^1/2，然后再应用幂法则。

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3. The Chain Rule | 链式法则

in rule is used when one function is nested inside another. If y = f(g(x)), then dy/dx = f'(g(x)) × g' ide function, leaving the inside unchanged, then multiply by the derivative of the inside.

链式法则用于一个函数嵌套在另一个函数内部的情况。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。用考试话语来说，先对外层函数求导，内层保持不变，再乘以内层函数的导数。

dy/dx = f'(g(x)) g'(x)

dy/du = 7u⁶ and du/dx = 6x. The chain rule gives dy/dx = 7(3x² − 5)⁶ × 6x = 42x(3x² − 5)⁶.

对于 y = (3x² − 5)⁷，设 u = 3x² − 5。则 y = u⁷，dy/du = 7u⁶，且 du/dx = 6x。链式法则给出 dy/dx = 7(3x² − 5)⁶ × 6x = 42x(3x² − 5)⁶。

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4. The Product Rule | 乘积法则

ust be used. If y = u v, where u and v are functions of x, then dy/dx = u dv/dx + v du/dx.

当一个函数写成两个较简单函数的乘积时，必须使用乘积法则。若 y = u v，其中 u 和 v 都是 x 的函数，则 dy/dx = u dv/dx + v du/dx。

dy/dx = u dv/dx + v du/dx

os x. Hence dy/dx = x² cos x + 2x sin x. This can also be written as x(x cos x + 2 sin x).

例如，若 y = x² sin x，选取 u = x² 和 v = sin x。则 du/dx = 2x，dv/dx = cos x。因此 dy/dx = x² cos x + 2x sin x。这也可以写成 x(x cos x + 2 sin x)。

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5. The Quotient Rule | 商法则

− u dv/dx) / v². The numerator order is important: start with v times the derivative of u.

当一个函数除以另一个函数时，需要使用商法则。若 y = u/v，其中 u 和 v 都是 x 的函数，则 dy/dx = (v du/dx − u dv/dx) / v²。分子的顺序很重要：从 v 乘以 u 的导数开始。

dy/dx = (v du/dx − u dv/dx) / v²

t rule gives dy/dx = [(x² − 3)(1) − (x + 1)(2x)] / (x² − 3)² = (−x² − 2x − 3) / (x² − 3)².

对于 y = (x + 1)/(x² − 3)，设 u = x + 1 和 v = x² − 3。则 du/dx = 1，dv/dx = 2x。商法则给出 dy/dx = [(x² − 3)(1) − (x + 1)(2x)] / (x² − 3)² = (−x² − 2x − 3) / (x² − 3)²。

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6. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分

The exponential function eˣ is special because its derivative is itself. More generally, if y = eᵏˣ, then dy/dx = k eᵏˣ. This follows directly from the chain rule, because the derivative of kx is k.

指数函数 eˣ 的特殊之处在于它的导数就是它本身。更一般地，若 y = eᵏˣ，则 dy/dx = k eᵏˣ。这直接由链式法则得出，因为 kx 的导数为 k。

d/dx (eᵏˣ) = k eᵏˣ

kx) = k / (kx) = 1/x. This surprising result is often tested in multiple-choice questions.

对于自然对数，ln x 的导数为 1/x。当自变量是 x 的倍数，例如 ln(kx)，链式法则给出 d/dx ln(kx) = k / (kx) = 1/x。这一令人意外的结果经常出现在选择题中。

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7. Differentiating Trigonometric Functions | 三角函数的微分

The three basic trigonometric derivatives must be memorised. For linear arguments such as sin(ax), cos(ax) and tan(ax), the chain rule introduces an extra factor of a in each case.

三个基本三角函数导数必须牢记。对于线性自变量如 sin(ax)、cos(ax) 和 tan(ax)，链式法则在每种情况下都会引入额外的因子 a。

• d/dx sin x = cos x, so d/dx sin(ax) = a cos(ax)
• d/dx cos x = −sin x, so d/dx cos(ax) = −a sin(ax)
• d/dx tan x = sec² x, so d/dx tan(ax) = a sec²(ax)

e sign in the derivative of cos x is a common source of exam errors, so always double-check your sign.

例如，y = sin(3x) 的导数为 dy/dx = 3 cos(3x)。cos x 导数中的负号是考试中常见的错误来源，因此务必再次检查符号。

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8. Implicit Differentiation | 隐函数微分

When y is not written explicitly as a function of x, you can still differentiate both sides of an equation with respect to x. Every time you differentiate a term involving y, you must multiply by dy/dx. This technique is called implicit differentiation.

当 y 没有明确写成 x 的函数时，你仍然可以对等式两边关于 x 进行微分。每次对含有 y 的项求导时，都必须乘以 dy/dx。这种技巧称为隐函数微分。

Rearranging gives dy/dx = −x/y. At the point (3, 4), the gradient of the tangent is −3/4.

例如，若 x² + y² = 25，对两边求导：2x + 2y dy/dx = 0。整理得到 dy/dx = −x/y。在点 (3, 4) 处，切线斜率为 −3/4。

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9. Parametric Differentiation | 参数微分

dx/dt, then divide. The formula is dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero.

曲线可以用参数 t 来定义，其中 x = f(t)，y = g(t)。要求 dy/dx，先求 dy/dt 和 dx/dt，然后相除。公式为 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt 不为零。

dy/dx = (dy/dt) / (dx/dt)

efore dy/dx = 6t² / 6t = t. This gives the gradient of the curve at any parameter value t.

例如，若 x = 3t² + 1 且 y = 2t³，则

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