A-Level Edexcel Maths: Differentiation Techniques and Applications | 爱德思A-Level数学：微分方法与应用

📚 A-Level Edexcel Maths: Differentiation Techniques and Applications | 爱德思A-Level数学：微分方法与应用

Differentiation is one of the largest topics in the Edexcel A-Level Mathematics specification, appearing in pure mathematics, mechanics and problem-solving questions. A strong command of derivative rules allows you to tackle gradient calculations, optimisation problems, tangent equations and rate-of-change questions with confidence.

微分是爱德思 A-Level 数学大纲中最大的主题之一，出现在纯数学、力学和问题解决题中。熟练掌握导数法则可以让你自信地处理斜率计算、优化问题、切线方程和变化率问题。

1. First Principles of Differentiation | 第一原理求导

The derivative of a function f(x) at a point x is defined as the limit of the gradient of a chord as the two points on the curve approach each other. In Edexcel papers, you may be asked to use this definition to differentiate simple functions such as f(x) = x².

函数 f(x) 在点 x 处的导数定义为曲线上两点逐渐靠近时弦斜率的极限。爱德思考试中可能要求用该定义求简单函数的导数，如 f(x) = x²。

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. Subtracting f(x) and dividing by h gives [2xh + h²]/h = 2x + h. As h tends to 0, the limit is 2x, so f'(x) = 2x.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。减去 f(x) 后除以 h 得到 [2xh + h²]/h = 2x + h。当 h 趋于 0 时，极限为 2x，因此 f'(x) = 2x。

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2. Power Rule and Term-by-Term Differentiation | 幂法则与逐项求导

If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹ for any real value of n. This rule works for positive, negative and fractional powers, so expressions like 1/x and √x must be rewritten as x⁻¹ and x^½ before differentiating.

若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹ 对任意实数 n 成立。该法则适用于正指数、负指数和分数指数，因此像 1/x 和 √x 这样的表达式必须先改写为 x⁻¹ 和 x^½ 再求导。

d/dx [xⁿ] = n xⁿ⁻¹

Example: find d/dx (3x⁴ − 5/x + 2√x). Rewrite the expression as 3x⁴ − 5x⁻¹ + 2x^½, then differentiate term by term to obtain 12x³ + 5x⁻² + x^(−½).

例：求 d/dx (3x⁴ − 5/x + 2√x)。将表达式改写为 3x⁴ − 5x⁻¹ + 2x^½，再逐项求导得到 12x³ + 5x⁻² + x^(−½)。

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3. Standard Derivatives | 常见函数求导

For Edexcel A-Level Mathematics, you are expected to know the derivatives of common functions by heart. The table below summarises the key results that appear most frequently in exam questions.

在爱德思 A-Level 数学中，你必须熟记常见函数的导数。下表总结了考题中最常出现的关键结果。

Function | 函数	Derivative | 导数
xⁿ	n xⁿ⁻¹
sin x	cos x
cos x	−sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x
aˣ	aˣ ln a
sin kx	k cos kx
cos kx	−k sin kx
e^(kx)	k e^(kx)

Learning these results accurately saves time and reduces sign errors. For example, the derivative of cos 3x is −3 sin 3x, not −sin 3x.

准确记住这些结果可以节省时间并减少符号错误。例如，cos 3x 的导数是 −3 sin 3x，而不是 −sin 3x。

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4. Product Rule | 乘法法则

When a function is the product of two simpler functions, use the product rule. If y = u v, where u and v are both functions of x, then the derivative is given by the following formula.

当一个函数是两个更简单函数的乘积时，使用乘法法则。若 y = u v，其中 u 和 v 都是 x 的函数，则导数由以下公式给出。

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

Example: differentiate y = x² eˣ. Let u = x² and v = eˣ, so du/dx = 2x and dv/dx = eˣ. The product rule gives dy/dx = x² eˣ + 2x eˣ = eˣ(x² + 2x).

例：对 y = x² eˣ 求导。设 u = x²，v = eˣ，则 du/dx = 2x，dv/dx = eˣ。由乘法法则得 dy/dx = x² eˣ + 2x eˣ = eˣ(x² + 2x)。

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5. Quotient Rule | 除法法则

For a function written as one expression divided by another, the quotient rule must be used. If y = u/v, then the derivative is (v du/dx − u dv/dx) / v². The order in the numerator is important: begin with v times du/dx.

对于写成两个表达式之商的函数，必须使用除法法则。若 y = u/v，则导数为 (v du/dx − u dv/dx) / v²。分子中的顺序很重要：先写 v 乘以 du/dx。

dy/dx = (v du/dx − u dv/dx) / v²

Example: differentiate y = x/(x+1). Let u = x and v = x+1, then du/dx = 1 and dv/dx = 1. Substituting into the formula gives dy/dx = [(x+1)(1) − x(1)] / (x+1)² = 1/(x+1)².

例：对 y = x/(x+1) 求导。设 u = x，v = x+1，则 du/dx = 1，dv/dx = 1。代入公式得 dy/dx = [(x+1)(1) − x(1)] / (x+1)² = 1/(x+1)²。

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6. Chain Rule and Composite Functions | 链式法则与复合函数

The chain rule is used when one function is inside another. If y = f(g(x)), let u = g(x), so y = f(u). The derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.

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))，令 u = g(x)，则 y = f(u)。y 对 x 的导数等于 y 对 u 的导数乘以 u 对 x 的导数。

dy/dx = dy/du × du/dx

Example: differentiate y = (2x+3)⁵. Let u = 2x+3, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 2. Therefore dy/dx = 5(2x+3)⁴ × 2 = 10(2x+3)⁴.

例：对 y = (2x+3)⁵ 求导。令 u = 2x+3，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 2。因此 dy/dx = 5(2x+3)⁴ × 2 = 10(2x+3)⁴。

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7. Parametric Differentiation | 参数方程求导

When a curve is defined by parametric equations x = f(t) and y = g(t), the gradient of the curve is found by dividing the derivative of y with respect to t by the derivative of x with respect to t.

当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，曲线斜率等于 y 对 t 的导数除以 x 对 t 的导数。

dy/dx = (dy/dt) / (dx/dt)

Example: a curve has x = t² and y = t³. Then dx/dt = 2t and dy/dt = 3t². The gradient is dy/dx = 3t² / 2t = 3t/2, provided t ≠ 0.

例：曲线参数方程为 x = t²，y = t³。则 dx/dt = 2t，dy/dt = 3t²。斜率为 dy/dx = 3t² / 2t = 3t/2，其中 t ≠ 0。

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