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。

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

The derivative of a function f(x) gives the gradient of the tangent to the curve at any point. It is formally defined from first principles as the limit of the difference 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

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h². The difference quotient simplifies 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。

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'(x) = n xⁿ⁻¹. This rule can be applied to sums, differences 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。

3. The Chain Rule | 链式法则

The chain rule is used when one function is nested inside another. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). In exam language, differentiate the outside 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)

For y = (3x² − 5)⁷, let u = 3x² − 5. Then y = u⁷, 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。

4. The Product Rule | 乘积法则

When a function is written as the product of two simpler functions, the product rule must 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

For example, if y = x² sin x, choose u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos 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)。

5. The Quotient Rule | 商法则

The quotient rule is needed when one function is divided by another. If y = u/v, where u and v are functions of x, then dy/dx = (v du/dx − 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

For y = (x + 1)/(x² − 3), let u = x + 1 and v = x² − 3. Then du/dx = 1 and dv/dx = 2x. The quotient 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。

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

For natural logarithms, the derivative of ln x is 1/x. When the argument is a multiple of x, such as ln(kx), the chain rule gives d/dx ln(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。

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)

For example, the derivative of y = sin(3x) is dy/dx = 3 cos(3x). The negative 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。

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。

For example, if x² + y² = 25, differentiate both sides: 2x + 2y dy/dx = 0. 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。

9. Parametric Differentiation | 参数微分

A curve can be defined using a parameter t, with x = f(t) and y = g(t). To find dy/dx, first find dy/dt and 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)

For instance, if x = 3t² + 1 and y = 2t³, then dx/dt = 6t and dy/dt = 6t². Therefore 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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Differentiation Techniques for Edexcel A-Level Mathematics | 爱德思A-Level数学微分技巧精讲