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

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

Differentiation is one of the central pillars of Edexcel A Level Pure Mathematics. It measures instantaneous rates of change and underpins topics such as curve sketching, optimisation, kinematics and modelling. A secure grasp of the basic rules, combined with the ability to apply them to products, quotients, composite functions, implicit relations and parametric curves, is essential for high marks on both Pure 1 and Pure 2 papers.

微分是爱德思 A Level 纯数学的核心支柱之一。它度量瞬时变化率，并为曲线草图、最优化、运动学和建模等主题奠定基础。牢固掌握基本求导法则，并能将其应用于乘积、商、复合函数、隐式关系与参数曲线，对于在 Pure 1 和 Pure 2 试卷中取得高分至关重要。

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

The derivative from first principles is defined by taking the limit of the average rate of change as the interval tends to zero. This limit gives the gradient of the chord as two points on the curve move infinitely close together.

第一性原理导数通过让平均变化率的区间趋于零来定义。当曲线上两点无限接近时，这个极限就是割线的斜率。

f'(x) = lim (h→0) [f(x+h) − f(x)] / h

To use this definition, form the difference quotient f(x+h) − f(x) divided by h, then simplify and let h approach zero. This process is often examined as a short proof in Edexcel Pure Mathematics.

使用该定义时，先构造差商 f(x+h) − f(x) 除以 h，然后化简并令 h 趋于零。这个过程在爱德思纯数学考试中常作为简短证明出现。

Worked example: for f(x) = x², the difference quotient is [(x+h)² − x²] / h = (2xh + h²) / h = 2x + h. As h→0, the limit is 2x, so f'(x) = 2x.

示例：对于 f(x) = x²，差商为 [(x+h)² − x²] / h = (2xh + h²) / h = 2x + h。当 h→0 时，极限为 2x，因此 f'(x) = 2x。

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2. Power, Constant and Sum Rules | 幂函数、常数与加法法则

For any real power n, the derivative of xⁿ is n times x to the power n minus one. Constants differentiate to zero, and derivatives of sums or differences can be taken term by term.

对于任意实数幂 n，xⁿ 的导数是 n 乘以 x 的 n−1 次幂。常数求导为零，和或差的导数可以逐项求导。

d/dx (xⁿ) = n xⁿ⁻¹   |   d/dx (c) = 0   |   d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Example: d/dx (5x³ − 2x + 7) = 15x² − 2. The constant 7 disappears because a horizontal shift does not change the gradient.

示例：d/dx (5x³ − 2x + 7) = 15x² − 2。常数 7 消失，因为水平平移不会改变斜率。

This rule is the foundation for all polynomial differentiation. In Edexcel exams it is often combined with factorisation before finding stationary points.

该法则是所有多项式微分的基础。在爱德思考试中，它常与因式分解结合，用于寻找驻点。

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3. Chain Rule | 链式法则

For y = f(u) and u = g(x), the chain rule states dy/dx = dy/du × du/dx. Think of the derivative as flowing from y to u, then from u to x.

对于 y = f(u) 和 u = g(x)，链式法则为 dy/dx = dy/du × du/dx。可以理解为导数先从 y 传到 u，再从 u 传到 x。

dy/dx = dy/du × du/dx

Example: y = (3x²+1)⁵. Set u = 3x²+1, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x. Hence dy/dx = 5u⁴ × 6x = 30x(3x²+1)⁴.

示例：y = (3x²+1)⁵。令 u = 3x²+1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x。因此 dy/dx = 5u⁴ × 6x = 30x(3x²+1)⁴。

The chain rule is also used in reverse in integration by substitution, so practising it thoroughly will benefit later topics.

链式法则在代换积分中也会逆向使用，因此充分练习对后续主题很有帮助。

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4. Product and Quotient Rules | 乘积法则与商法则

When differentiating the product of two functions, keep one fixed and differentiate the other, then swap. The product rule is d/dx (uv) = u dv/dx + v du/dx.

对两个函数的乘积求导时，先固定一个函数并求另一个的导数，然后交换。乘积法则为 d/dx (uv) = u dv/dx + v du/dx。

d/dx (uv) = u dv/dx + v du/dx

Example: y = x² eˣ. Let u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, so dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x).

示例：y = x² eˣ。令 u = x²，v = eˣ。则 u’ = 2x，v’ = eˣ，所以 dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x)。

For a quotient, the order matters because subtraction is not commutative. The quotient rule is d/dx (u/v) = (v du/dx − u dv/dx) / v².

对于商函数，顺序非常重要，因为减法不满足交换律。商法则为 d/dx (u/v) = (v du/dx − u dv/dx) / v²。

d/dx (u/v) = (v du/dx − u dv/dx) / v²

Example: y = x²/(x+1). Let u = x² and v = x+1. Then du/dx = 2x and dv/dx = 1, so dy/dx = [(x+1)(2x) − x²(1)] / (x+1)² = (x² + 2x) / (x+1)².

示例：y = x²/(x+1)。令 u = x²，v = x+1。则 du/dx = 2x，dv/dx = 1，所以 dy/dx = [(x+1)(2x) − x²(1)] / (x+1)² = (x² + 2x) / (x+1)²。

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5. Differentiating Exponentials and Logarithms | 指数函数与对数函数微分

The exponential function eˣ is unique because its derivative is itself. For a general base a, multiply by ln a. The natural logarithm differentiates to the reciprocal of x.

指数函数 eˣ 的特殊之处在于它的导数就是它自身。对于一般底数 a，需要乘以 ln a。自然对数的导数是 x 的倒数。

d/dx (eˣ) = eˣ   |   d/dx (eᵏˣ) = k eᵏˣ   |   d/dx (aˣ) = aˣ ln a   |   d/dx (ln x) = 1/x

Example: y = 3e²ˣ + 4 ln 5x. Using the chain rule, y’ = 3 × 2e²ˣ + 4 × (1/x) = 6e²ˣ + 4/x. The logarithm of

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