Differentiation and Its Applications for Edexcel A-Level Maths | Edexcel A-Level 数学：微分法及其应用

📚 Differentiation and Its Applications for Edexcel A-Level Maths | Edexcel A-Level 数学：微分法及其应用

Differentiation is one of the most important topics in the Edexcel A-Level Mathematics specification. It appears in AS and A-Level papers, both as stand-alone questions and within applied problems such as optimisation, rates of change and curve sketching. A confident command of derivative rules and their interpretations is essential for success in both Pure Mathematics and Mechanics.

微分法是 Edexcel A-Level 数学考试中最核心的主题之一。它在 AS 和 A-Level 试卷中既会单独出现，也会内嵌于最优化、变化率和曲线作图等应用问题中。熟练掌握导数运算规则及其几何与物理意义，是学好纯数学与力学部分的关键。

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1. First Principles Differentiation | 从第一原理理解导数

The derivative of a function f(x) at a point is defined as the limit of the average rate of change as the interval h tends to zero. In Edexcel exam papers, you are often asked to use first principles to find the derivative of simple polynomials such as x² or x³.

函数 f(x) 在一点的导数定义为当增量 h 趋于零时平均变化率的极限。在 Edexcel 试卷中，常会要求利用第一原理求简单多项式如 x² 或 x³ 的导数。

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

To apply the definition, expand f(x+h), simplify the numerator, cancel h, and then substitute h = 0. This method is particularly useful for demonstrating understanding of why the derivative gives the gradient of a tangent.

使用该定义时，先展开 f(x+h)，化简分子并约去 h，再代入 h = 0。这一方法特别有助于理解导数为何表示切线的斜率。

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h². Subtracting f(x) gives 2xh + h², and dividing by h gives 2x + h. As h tends to zero, the limit is 2x.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。减去 f(x) 得到 2xh + h²，再除以 h 得到 2x + h。当 h 趋于零时，极限为 2x。

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2. Basic Rules of Differentiation | 基本微分法则

Edexcel expects you to differentiate polynomial, rational and radical functions fluently using the power rule. For any real power n, the derivative of xⁿ is n xⁿ⁻¹.

Edexcel 要求考生能运用幂法则熟练地对多项式、有理函数和根式函数求导。对于任意实数指数 n，xⁿ 的导数为 n xⁿ⁻¹。

d/dx (xⁿ) = n xⁿ⁻¹

You should also remember the constant multiple rule and the sum rule: differentiation is linear. For example, the derivative of 5x⁴ − 3x² + 2x − 7 is 20x³ − 6x + 2.

你还需要记住常数倍法则与加法法则：微分是线性运算。例如，5x⁴ − 3x² + 2x − 7 的导数为 20x³ − 6x + 2。

When dealing with square roots or reciprocals, rewrite them as powers of x first. For instance, √x becomes x^(1/2) and 1/x² becomes x⁻², so the power rule can be applied directly.

处理平方根或倒数时，先把它们改写为 x 的幂。例如，√x 写成 x^(1/2)，1/x² 写成 x⁻²，这样就可以直接使用幂法则。

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3. Product and Quotient Rules | 乘法法则与商法则

When differentiating a product of two functions, use the product rule: multiply the derivative of the first function by the second, then add the first function multiplied by the derivative of the second.

当对两个函数的乘积求导时，使用乘法法则：第一个函数的导数乘以第二个函数，再加上第一个函数乘以第二个函数的导数。

d/dx (uv) = u’v + uv’

For a quotient, the derivative is given by (u’v − uv’) / v². Always identify u and v clearly before differentiating, and be careful with signs in the numerator of the quotient rule.

对于两个函数的商，其导数为 (u’v − uv’) / v²。求导前应明确 u 和 v 的表达式，并特别注意商法则分子中的符号。

A typical Edexcel question may ask you to differentiate y = x² sin x. Here u = x² and v = sin x, so y’ = 2x sin x + x² cos x. With quotients, avoid expanding the denominator unless it simplifies the work.

一道典型的 Edexcel 题目可能要求对 y = x² sin x 求导。这里 u = x²，v = sin x，因此 y’ = 2x sin x + x² cos x。对于商式，除非能化简，否则不要展开分母。

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4. Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In A-Level questions, you can also write u = g(x), find dy/du and du/dx, then multiply them.

链式法则用于复合函数求导。若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。在 A-Level 题目中，也可以设 u = g(x)，先求 dy/du 与 du/dx，再将两者相乘。

dy/dx = dy/du × du/dx

For example, to differentiate y = (3x² + 1)⁵, let u = 3x² + 1. Then dy/du = 5u⁴ and du/dx = 6x, so dy/dx = 30x(3x² + 1)⁴.

例如，要对 y = (3x² + 1)⁵ 求导，设 u = 3x² + 1。则 dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 30x(3x² + 1)⁴。

The chain rule is often combined with trigonometric, exponential and logarithmic functions. Always remember to multiply by the derivative of the inner function, because forgetting this step is one of the most frequent errors in A-Level differentiation.

链式法则常与三角函数、指数函数和对数函数结合使用。务必记住乘以内层函数的导数，因为遗漏这一步是 A-Level 微分中最常见的错误之一。

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5. Differentiating Standard Functions | 标准函数的导数

Memorising the derivatives of standard functions is essential. The derivative of sin x is cos x, the derivative of cos x is −sin x, and the derivative of tan x is sec² x. For exponentials, d/dx(eˣ) = eˣ and d/dx(aˣ) = aˣ ln a.

熟记标准函数的导数至关重要。sin x 的导数为 cos x，cos x 的导数为 −sin x，tan x 的导数为 sec² x。对于指数函数，d/dx(eˣ) = eˣ，d/dx(aˣ) = a

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