Edexcel A-Level Maths: Differentiation and Applications | 爱德思A-Level数学：微分及其应用

📚 Edexcel A-Level Maths: Differentiation and Applications | 爱德思A-Level数学：微分及其应用

In A-Level Edexcel Mathematics, differentiation is one of the most powerful tools in pure mathematics. It measures how a function changes as its input changes, giving the gradient of a curve at any point. This article covers key differentiation techniques and their applications, preparing students for Paper 1 and Paper 2 topics.

在爱德思A-Level数学中，微分是纯数学最重要的工具之一。它衡量函数随自变量变化的变化率，给出曲线上任意点的斜率。本文涵盖核心微分技巧及其应用，帮助学生备考纯数学1和纯数学2。

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1. Differentiation from First Principles | 从第一性原理求导

Differentiation from first principles gives the formal definition of a derivative. For a function f, the derivative at a point x is the limit of the gradient of a chord as the interval h tends to zero:

从第一性原理求导给出导数的正式定义。对于函数 f，点 x 处的导数是当区间 h 趋于零时割线斜率的极限：

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

important to show the expansion and simplification steps clearly before taking the limit.

爱德思考试常要求考生对简单多项式如 f(x)=x² 或 f(x)=x³ 应用该定义。必须清晰展示展开和化简步骤，再取极限。

) = (x+h)² − x² = 2xh + h². Dividing by h gives 2x + h, and as h → 0 the derivative is 2x.

例：对于 f(x)=x²，f(x+h)−f(x)=(x+h)²−x²=2xh+h²。除以 h 得 2x+h，当 h→0 时导数为 2x。

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2. Basic Derivative Rules | 基本求导法则

xⁿ⁻¹. Constants differentiate to zero, and sums and differences are differentiated term by term. Edexcel also expects known derivatives of standard functions, summarised below.

一旦建立了定义，标准法则可使求导高效。若 y=axⁿ，则 dy/dx=naxⁿ⁻¹。常数导数为零，和与差可逐项求导。爱德思还要求掌握常见函数的导数，汇总如下。

Function f(x)	Derivative f'(x)
xⁿ	n xⁿ⁻¹
sin x	cos x
cos x	−sin x
eˣ	eˣ
ln x	1/x

These results are used constantly in later sections, especially when combined with the chain, product and quotient rules.

这些结果在后续内容中频繁使用，尤其是在结合链式法则、乘法法则和除法法则时。

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3. Chain Rule | 链式法则

f y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. This can also be written as dy/dx = f'(g(x)) × g'(x).

复合函数求导需要链式法则。若 y=f(u) 且 u=g(x)，则 dy/dx=dy/du × du/dx。也可写作 dy/dx=f'(g(x)) × g'(x)。

e y = (3x² + 5)⁷, set u = 3x² + 5. Then dy/du = 7u⁶ = 7(3x²+5)⁶ and du/dx = 6x. Therefore:

例如，对 y=(3x²+5)⁷ 求导，令 u=3x²+5。则 dy/du=7u⁶=7(3x²+5)⁶，du/dx=6x。因此：

dy/dx = 7(3x²+5)⁶ × 6x = 42x(3x²+5)⁶

Common chain rule cases include y = sin(2x), y = e^x² and y = ln(5x−1). Always identify the inner function u before differentiating.

常见链式法则情形包括 y=sin(2x)、y=e^x² 和 y=ln(5x−1)。求导前务必先识别内层函数 u。

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4. Product Rule | 乘法法则

When two functions multiply, the product rule states:

当两个函数相乘时，乘法法则为：

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

Here u and v are both functions of x. Example: differentiate y = x²eˣ. Take u = x², v = eˣ, du/dx = 2x, dv/dx = eˣ. Then:

其中 u 和 v 均为 x 的函数。例：对 y=x²eˣ 求导，取 u=x²，v=eˣ，du/dx=2x，dv/dx=eˣ。则：

dy/dx = x²eˣ + 2x eˣ = x eˣ(x+2)

Product rule is useful for expressions such as x sin x, (x+1) ln x, and x² cos x. Make sure to keep the original u and v in the correct positions.

乘法法则适用于 x sin x、(x+1)ln x、x² cos x 等表达式。注意保持 u 和 v 在正确位置。

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5. Quotient Rule | 除法法则

If y = u / v, where u and v are functions of x, then the quotient rule states:

若 y=u/v，其中 u 和 v 是 x 的函数，则除法法则为：

dy/dx = (v du/dx − u dv/dx) / v²

rentiate y = x / (x+1). Let u = x, v = x+1. Then du/dx = 1, dv/dx = 1. Substituting gives:

例：对 y=x/(x+1) 求导。令 u=x，v=x+1，则 du/dx=1，dv/dx=1。代入得：

dy/dx = [(x+1)(1) − x(1)] / (x+1)² = 1 / (x+1)²

Remember that the numerator order is v du/dx first, then minus u dv/dx. The denominator is always v².

记住分子顺序是先 v·du/dx，再减去 u·dv/dx。分母始终为 v²。

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6. Tangents and Normals | 切线与法线

The gradient of the tangent at x = a is m = f'(a). The tangent equation is:

在 x=a 处切线斜率为 m=f'(a)。切线方程为：

y − f(a) = m (x − a)

mₙ = −1/m, provided m ≠ 0. If m = 0, the tangent is horizontal and the normal is vertical.

法线与切线垂直，故其斜率 mₙ=−1/m（当 m≠0）。若 m=0，切线水平，法线竖直。

Example: for y = x² at x = 1, f' m = 2 and f(1)=1. Tangent: y−1=2(x−1). Normal gradient is −1/2, so normal: y−1 = −½(x−1).

例：对于 y=x² 在 x=1 处，f'(x)=2x，故 m=2，f(1)=1。切线：y−1=2(x−1)。法线斜率为 −1/2，所以法线：y−1=−½(x−1)。

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7. Stationary Points and Turning Points | 驻点与驻点性质

, local minimum, or stationary point of inflection, use the second derivative d²y/dx²: if d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, the point is a local maximum. If d

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