Edexcel A Level Maths: Differentiation Techniques and Applications | 爱德思A Level数学：微分技巧与应用

📚 Edexcel A Level Maths: Differentiation Techniques and Applications | 爱德思A Level数学：微分技巧与应用

Differentiation is one of the most important topics in Edexcel A Level Mathematics. It appears in Pure Mathematics units from AS to A2 and forms the basis for many applications in gradients, optimisation, kinematics and rates of change. This article reviews the key techniques, rules and exam strategies you need to master differentiation with confidence.

微分是爱德思A Level数学中最重要的主题之一。它出现在从AS到A2的纯数学单元中，并且是梯度、优化、运动学和变化率等许多应用的基础。本文回顾你需要掌握的关键技巧、法则和考试策略，帮助你自信地掌握微分。

1. Gradient of a Curve and First Principles | 曲线斜率与第一性原理

The derivative of a function y = f(x) gives the gradient of the tangent to the curve at any point. From first principles, this gradient is found by taking the limit of the gradient of a chord joining two points on the curve. The formal definition is written as f'(x) = lim(h→0) [f(x+h) – f(x)] / h.

函数 y = f(x) 的导数给出了曲线上任意一点切线的斜率。根据第一性原理，这个斜率通过取曲线上两点之间弦的斜率的极限来求得。其正式定义写作 f'(x) = lim(h→0) [f(x+h) – f(x)] / h。

f'(x) = lim(h→0) [f(x+h) – f(x)] / h

For example, if f(x) = x², the first principles calculation gives f'(x) = 2x. This result is one of the most frequently used building blocks in A Level differentiation.

例如，如果 f(x) = x²，第一性原理的计算会得到 f'(x) = 2x。这个结果是A Level微分中最常用的基础公式之一。

2. Basic Differentiation Rules and Common Derivatives | 基本求导法则与常见导数

For any power function y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. This rule can be applied term by term to polynomials. If y = axⁿ + bxᵐ, then dy/dx = anxⁿ⁻¹ + bmxᵐ⁻¹, where a and b are constants.

对于任意幂函数 y = xⁿ，其导数为 dy/dx = nxⁿ⁻¹。这个法则可以逐项应用于多项式。如果 y = axⁿ + bxᵐ，那么 dy/dx = anxⁿ⁻¹ + bmxᵐ⁻¹，其中 a 和 b 是常数。

d/dx(xⁿ) = nxⁿ⁻¹

You should also memorise the derivatives of key functions: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(eˣ) = eˣ, and d/dx(ln x) = 1/x. These are essential for later topics such as trigonometric differentiation and exponential models.

你还应该熟记关键函数的导数：d/dx(sin x) = cos x、d/dx(cos x) = -sin x、d/dx(eˣ) = eˣ、以及 d/dx(ln x) = 1/x。这些对于后续的三角微分和指数模型等内容至关重要。

3. Chain Rule for Composite Functions | 复合函数的链式法则

The chain rule is used to differentiate composite functions of the form y = f(g(x)). If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. This is sometimes written as dy/dx = f'(g(x)) × g'(x).

链式法则用于对形如 y = f(g(x)) 的复合函数求导。如果 y = f(u) 且 u = g(x)，那么 dy/dx = dy/du × du/dx。这有时也写作 dy/dx = f'(g(x)) × g'(x)。

dy/dx = dy/du × du/dx

A typical example is y = (3x² + 5)⁴. Let u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 24x(3x² + 5)³. The chain rule is especially useful when expressions are raised to powers.

一个典型例子是 y = (3x² + 5)⁴。设 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³ 且 du/dx = 6x，得到 dy/dx = 24x(3x² + 5)³。当表达式带有幂次时，链式法则尤其有用。

4. Product Rule | 乘法法则

When a function is the product of two simpler functions, y = u(x)v(x), the derivative is not simply the product of the derivatives. Instead, use the product rule: dy/dx = u dv/dx + v du/dx. This can be remembered as “first times derivative of second plus second times derivative of first”.

当一个函数是两个较简单函数的乘积时，即 y = u(x)v(x)，其导数并不是两个导数的简单乘积。此时应使用乘法法则：dy/dx = u dv/dx + v du/dx。这可以记为“第一个乘以第二个的导数，再加上第二个乘以第一个的导数”。

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

For instance, if y = x² sin x, let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x, so dy/dx = x² cos x + 2x sin x. Product rule questions often appear in the second half of Edexcel Pure papers and require careful simplification.

例如，如果 y = x² sin x，设 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，所以 dy/dx = x² cos x + 2x sin x。乘法法则的题目经常出现在爱德思纯数学试卷的后半部分，并且需要仔细化简。

5. Quotient Rule | 除法法则

For rational functions y = u(x)/v(x), the quotient rule states that dy/dx = (v du/dx – u dv/dx) / v². The order in the numerator is important: it is “bottom times derivative of top minus top times derivative of bottom”, all divided by the square of the bottom.

对于有理函数 y = u(x)/v(x)，除法法则指出 dy/dx = (v du/dx – u dv/dx) / v²。分子中的顺序很重要：应为“分母乘以分子的导数，减去分子乘以分母的导数”，再除以分母的平方。

dy/dx = (v du/dx – u dv/dx) / v²

As an example, differentiate y = x / (x² + 1). Let u = x and v = x² + 1. Then du/dx = 1 and dv/dx = 2x. The derivative is dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)².

例如，对 y = x / (x² + 1) 求导。设 u = x，v = x² + 1。则 du/dx = 1，dv/dx = 2x。其导数为 dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)²。

6. Differentiating Trigonometric, Exponential and Logarithmic Functions | 三角函数、指数函数与对数函数的微分

Edexcel A Level Mathematics requires fluency with the derivatives of sin, cos, tan, eˣ and ln x. The basic results are: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, and d/dx(tan x) = sec² x. For exponential functions, d/dx(eˣ) = eˣ, and more generally d/dx(eᵏˣ) = keᵏˣ.

爱德思A Level数学要求熟练掌握 sin、cos、tan、eˣ 和 ln x 的导数。基本结果是：d/dx(sin x) = cos x、d/dx(cos x) = -sin x、以及 d/dx(tan x) = sec² x。对于指数函数，d/dx(eˣ) = eˣ，更一般地，d/dx(eᵏˣ) = keᵏˣ。

d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(tan x) = sec² x

For composite versions, combine these with the chain rule. For example, d/dx(sin 5x) = 5 cos 5x and d/dx(e³ˣ) = 3e³ˣ. When differentiating ln(f(x)), the result is f'(x)/f(x), which follows from the chain rule.

对于复合形式，需要将这些公式与链式法则结合使用。例如，d/dx(sin 5x) = 5 cos 5x，d/dx(e³ˣ) = 3e³ˣ。在对 ln(f(x)) 求导时，结果是 f'(x)/f(x)，这可以从链式法则推出。

7. Parametric Differentiation | 参数方程微分

When a curve is defined by parametric equations x = x(t) and y = y(t), the gradient of the tangent is given by dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero. This formula allows you to find tangents and stationary points without eliminating the parameter.

当曲线由参数方程 x = x(t) 和 y = y(t) 定义时，切线的斜率由 dy/dx = (dy/dt) / (dx/dt) 给出，前提是 dx/dt 不为零。这个公式使你无需消去参数就能找到切线和驻点。

dy/dx = (dy/dt) ÷ (dx/dt)

For example, if x = t² + 1 and y = t³ – 2t, then dx/dt = 2t and dy/dt = 3t² – 2. Hence dy/dx = (3t² – 2) / 2t. A stationary point occurs when dy/dx = 0, which happens when 3t² – 2 = 0, so t = ±√(2/3).

例如，如果 x = t² + 1 且 y = t³ – 2t，那么 dx/dt = 2t，dy/dt = 3t² – 2。因此 dy/dx = (3t² – 2) / 2t。当 dy/dx = 0 时出现驻点，即 3t² – 2 = 0，所以 t = ±√(2/3)。

8. Implicit Differentiation | 隐函数微分

Implicit differentiation is used when y is not explicitly given as a function of x. Differentiate both sides of the equation with respect to x, treating y as a function of x. Whenever you differentiate a term involving y, multiply by dy/dx.

当 y 没有被显式地表示为 x 的函数时，就使用隐函数微分。对方程两边关于 x 求导，并将 y 视为 x 的函数。每当对含有 y 的项求导时，都要乘以 dy/dx。

d/dx(yⁿ) = nyⁿ⁻¹ × dy/dx

For instance, to differentiate x² + y² = 25, differentiate term by term: 2x + 2y(dy/dx) = 0, so dy/dx = -x/y. This gives the gradient of the tangent at any point on the circle without solving for y first.

例如，对 x² + y² = 25 求导，逐项得到：2x + 2y(dy/dx) = 0，所以 dy/dx = -x/y。这样就无需先解出 y 即可得到圆上任意一点切线的斜率。

9. Second Derivatives and Points of Inflection | 二阶导数与拐点

The second derivative, denoted d²y/dx² or f”(x), is the derivative of the first derivative. It describes the rate of change of the gradient and is used to determine concavity. If d²y/dx² > 0, the curve is concave upward; if d²y/dx² < 0, it is concave downward.

二阶导数记作 d²y/dx² 或 f”(x)，是一阶导数的导数。它描述斜率的变化率，并用于判断曲线的凹凸性。如果 d²y/dx² > 0，曲线向上凹；如果 d²y/dx² < 0，曲线向下凹。

A point of inflection occurs where the concavity changes sign. At such a point, the second derivative is usually zero, but this condition alone is not sufficient; you must check that d²y/dx² changes sign on either side.

拐点出现在凹凸性改变符号的位置。在这样的点上，二阶导数通常为零，但仅凭这个条件还不够；你必须检查 d²y/dx² 在该点两侧是否变号。

d²y/dx² = d/dx(dy/dx)

For example, if y = x³ – 3x² + 2, then dy/dx = 3x² – 6x and d²y/dx² = 6x – 6. Setting d²y/dx² = 0 gives x = 1. Since the second derivative changes from negative to positive around x = 1, there is a point of inflection at x = 1.

例如，如果 y = x³ – 3x² + 2，那么 dy/dx = 3x² – 6x，d²y/dx² = 6x – 6。令 d²y/dx² = 0 得到 x = 1。由于二阶导数在 x = 1 附近由负变正，因此在 x = 1 处存在一个拐点。

10. Tangents, Normals and Stationary Points | 切线、法线与驻点

Once dy/dx is known, the equation of the tangent at a point (a, b) is y – b = m(x – a), where m is the gradient of the tangent. The normal is perpendicular to the tangent, so its gradient is -1/m, provided m is not zero.

一旦知道 dy/dx，点 (a, b) 处的切线方程就是 y – b = m(x – a)，其中 m 是切线的斜率。法线垂直于切线，因此其斜率为 -1/m，前提是 m 不为零。

A stationary point occurs when dy/dx = 0. To classify it, use the second derivative test: if d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum; if d²y/dx² = 0, the test is inconclusive and you should examine the sign of dy/dx on either side.

当 dy/dx = 0 时出现驻点。要判断其类型，可使用二阶导数检验：如果 d²y/dx² > 0，该点是局部极小值；如果 d²y/dx² < 0，该点是局部极大值；如果 d²y/dx² = 0，该检验无法判断，此时应检查 dy/dx 在两侧的符号。

Tangent: y – b = m(x – a), Normal: y – b = (-1/m)(x – a)

These ideas are regularly tested in Edexcel Pure Mathematics, often in modelling or optimisation contexts where you must interpret the stationary point in the original problem.

这些思想在爱德思纯数学中经常考查，通常在建模或优化情境中，你需要将驻点解释回原始问题中。

11. Connected Rates of Change and Modelling | 相关变化率与建模

Connected rates of change problems link two or more changing quantities using the chain rule. For example, if the radius of a circle increases at a known rate and you want the rate of change of its area, use dA/dt = dA/dr × dr/dt.

相关变化率问题利用链式法则将两个或多个变化中的量联系起来。例如，如果圆的半径以已知速率增大，而你想要求面积的变化率，可以使用 dA/dt = dA/dr × dr/dt。

dA/dt = dA/dr × dr/dt

If A = πr², then dA/dr = 2πr. If dr/dt = 0.5 cm/s and r = 4 cm, then dA/dt = 2π × 4 × 0.5 = 4π cm²/s. Always state the units clearly and make sure you know which quantity is changing with respect to time.

如果 A = πr²，那么 dA/dr = 2πr。如果 dr/dt = 0.5 cm/s 且 r = 4 cm，则 dA/dt = 2π × 4 × 0.5 = 4π cm²/s。务必清楚地写出单位，并确保你知道哪个量在随时间变化。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

When answering differentiation questions, always show clear, logical working. Write down the rule you are using before substituting values, because Edexcel marking schemes often award method marks even if the final answer is incorrect.

在回答微分问题时，一定要写出清晰、有逻辑的过程。在代入数值之前先写出所使用的法则，因为爱德思的评分方案通常会授予方法分，即使最终答案不正确。

Common mistakes include forgetting to multiply by the derivative of the inside function when using the chain rule, incorrectly applying the quotient rule numerator order, and failing to simplify fully. Also check whether a question asks for the tangent or the normal, and whether the gradient is given or must be found.

常见错误包括：使用链式法则时忘记乘以内函数的导数；错误地应用除法法则的分子顺序；以及没有完全化简。还要检查题目问的是切线还是法线，以及斜率是已经给出还是需要自己求。

Always check the domain of the function, especially for ln x and rational functions.
For stationary points, confirm both dy/dx = 0 and the nature using the second derivative or sign change.
In connected rates, use the chain rule with respect to time t.
For parametric curves, dy/dx is in terms of the parameter, not x.

始终检查函数的定义域，尤其是 ln x 和有理函数。对于驻点，要确认 dy/dx = 0，并利用二阶导数或符号变化判断其性质。在相关变化率中，要用对时间 t 的链式法则。对于参数曲线，dy/dx 是用参数表示的，而不是用 x 表示的。

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Edexcel A Level Maths: Differentiation Techniques and Applications | 爱德思A Level数学：微分技巧与应用