Applications of More Complex Differentiation | 更复杂微分的应用

📚 Applications of More Complex Differentiation | 更复杂微分的应用

At A-level, differentiation moves beyond simple powers into composites, products, quotients, implicit curves and parametric motion. This article reviews the core rules and then applies them to the problem types most often tested in AQA Mathematics.

在 A-level 阶段，求导从简单幂函数扩展到复合函数、乘积、商、隐函数曲线和参数运动。本文回顾核心法则，并将其应用到 AQA 数学考试中最常考的题型。

1. Review of the Core Rules | 核心法则回顾

For more complex differentiation you must be fluent with the chain rule, product rule and quotient rule. The chain rule is used whenever one function is inside another: d/dx [f(g(x))] = f′(g(x)) × g′(x).

对于更复杂的求导，你必须熟练运用链式法则、乘积法则和商法则。只要一个函数嵌套在另一个函数内部，就要用到链式法则：d/dx [f(g(x))] = f′(g(x)) × g′(x)。

The product rule handles u(x)v(x): d/dx [uv] = u′v + uv′. The quotient rule handles u/v: d/dx [u/v] = (v u′ − u v′) / v².

乘积法则处理 u(x)v(x)：d/dx [uv] = u′v + uv′。商法则处理 u/v：d/dx [u/v] = (v u′ − u v′) / v²。

You should also recall the derivatives of standard functions: d/dx [eˣ] = eˣ, d/dx [ln x] = 1/x, d/dx [sin x] = cos x, d/dx [cos x] = −sin x, and d/dx [tan x] = sec² x.

你还应该记住标准函数的导数：d/dx [eˣ] = eˣ，d/dx [ln x] = 1/x，d/dx [sin x] = cos x，d/dx [cos x] = −sin x，以及 d/dx [tan x] = sec² x。