Differentiation and Integration: Core Techniques for Edexcel A-Level Maths | Edexcel A-Level数学微积分核心技巧

📚 Differentiation and Integration: Core Techniques for Edexcel A-Level Maths | Edexcel A-Level数学微积分核心技巧

This revision guide covers the essential differentiation and integration techniques required for the Edexcel A-Level Mathematics pure specification. You will need to apply these methods accurately in both structured and problem-solving questions, often in combination with coordinate geometry, trigonometry and algebra.

本复习指南涵盖Edexcel A-Level数学纯数考试中必需的微分与积分技巧。你需要在结构化题和应用题中准确使用这些方法，并经常与坐标几何、三角学及代数知识结合起来。

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1. First Principles and Derivative Rules | 第一性原理与求导法则

A-Level Edexcel Pure Mathematics requires you to define the derivative from first principles. The gradient of a chord between x and x + h is [f(x+h) − f(x)] / h. As h tends to 0, this gives the derivative f′(x).

Edexcel A-Level纯数学要求你从第一性原理定义导数。x与x+h之间弦的斜率是[f(x+h)−f(x)]/h。当h趋近于0时，就得到导数f′(x)。

f′(x) = lim(h→0) [f(x+h) − f(x)] / h

For polynomials, the power rule states that differentiating xⁿ gives n xⁿ⁻¹. Constants differentiate to zero, and the derivative of a sum is the sum of the derivatives.

对于多项式，幂法则表明xⁿ的导数是n xⁿ⁻¹。常数的导数为零，和的导数等于导数的和。

d/dx (xⁿ) = n xⁿ⁻¹, d/dx (c) = 0

Always simplify expressions such as √x or 1/x² into index form before differentiating. This reduces errors and makes the power rule straightforward.

在求导前始终将√x或1/x²等表达式化简为指数形式。这可以减少错误，使幂法则更加直接。

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2. Chain Rule | 链式法则

The chain rule is used when one function is inside another. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. This rule appears frequently in Edexcel exam questions involving composite functions such as (3x²+5)⁷ or sin(2x).

链式法则用于一个函数嵌套在另一个函数中的情况。如果y=f(u)且u=g(x)，则dy/dx=dy/du×du/dx。该法则在Edexcel考试中经常出现，涉及复合函数，如(3x²+5)⁷或sin(2x)。

dy/dx = dy/du × du/dx

With practice, you can differentiate composite functions in one line. For y = (2x+3)⁵, let u = 2x+3, so y = u⁵, dy/du = 5u⁴ and du/dx = 2, giving dy/dx = 10(2x+3)⁴.

通过练习，你可以一行写出复合函数的导数。对于y=(2x+3)⁵，设u=2x+3，则y=u⁵，dy/du=5u⁴，du/dx=2，因此dy/dx=10(2x+3)⁴。

For trigonometric functions, remember that derivative of sin x is cos x and derivative of cos x is −sin x. The chain rule then gives d/dx sin(ax+b) = a cos(ax+b).

对于三角函数，记住sin x的导数是cos x，cos x的导数是−sin x。链式法则给出d/dx sin(ax+b)=a cos(ax+b)。

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

When two functions are multiplied, the product rule must be used. If y = u v, then dy/dx = u dv/dx + v du/dx. Do not simply multiply the individual derivatives.

当两个函数相乘时，必须使用乘积法则。如果y=u v，则dy/dx=u dv/dx+v du/dx。不要简单地将两个导数相乘。

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

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

对于商y=u/v，商法则给出dy/dx=(v du/dx−u dv/dx)/v²。顺序很重要，因为减法不可交换。

d/dx (u/v) = (v du/dx − u dv/dx) / v²

Edexcel often asks for differentiation of rational functions such as y = x² sin x or y = eˣ / (x+1). Identify u and v clearly, find du/dx and dv/dx, then substitute carefully into the formula.

Edexcel常考有理函数或混合函数的求导，如y=x² sin x或y=eˣ/(x+1)。请清晰识别u和v，求出du/dx和dv/dx，然后小心代入公式。

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4. Implicit Differentiation | 隐函数求导

Some equations define y implicitly rather than as y = f(x). In implicit differentiation, differentiate every term with respect to x and multiply by dy/dx whenever differentiating a function of y.

有些方程隐式地定义y，而不是显式的y=f(x)。在隐函数求导中，对每一项关于x求导，每当对y的函数求导时乘以dy/dx。

d/dx (y²) = 2y dy/dx

For example, given x² + y² = 25, differentiating gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. This is useful for finding tangents to circles and other curves.

例如，对于x²+y²=25，求导得到2x+2y dy/dx=0，因此dy/dx=−x/y。这对于求圆和其他曲线的切线很有用。

2x + 2y dy/dx = 0 ⇒ dy/dx = −x/y

When simplifying, do not cancel terms involving dy/dx incorrectly. Collect all dy/dx terms on one side, factorise, and then solve.

化简时不要错误地约去含有dy/dx的项。将所有dy/dx项合并到一边，提取公因子后再求解。

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5. Parametric Differentiation | 参数方程求导

When x and y are given in terms of a parameter t, use dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero. This converts parametric equations into a Cartesian gradient.

当x和y用参数t表示时，使用dy/dx=(dy/dt)/(dx/dt)，前提是dx/dt不为零。这将参数方程转换为笛卡尔梯度。

dy/dx = (dy/dt) ÷ (dx/dt)

Example: x = t², y = 2t gives dx/dt = 2t and dy/dt = 2, so dy/dx = 1/t. You can then find the equation of a tangent at a specific t value.

示例：x=t²，y=2t，则dx/dt=2t，dy/dt=2，所以dy/dx=1/t。然后你可以求特定t值处的切线方程。

For second derivatives in parametric form, use d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) ÷ dx/dt. This is often tested as a higher-skill question.

参数形式的二阶导数使用d²y/dx²=d/dx(dy/dx)=d/dt(dy/dx

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