📚 Implicit Differentiation | 隐函数求导
Implicit differentiation is a key topic in Edexcel A-Level Mathematics. It is used when y cannot easily be written as a function of x, yet you still need to find the gradient dy/dx. This article explains the method, common examples, and exam techniques, with parallel English and Chinese explanations.
隐函数求导是 Edexcel A-Level 数学中的一个重要主题。当 y 不容易写成 x 的函数，但你仍需要求梯度 dy/dx 时，就会用到它。本文通过中英双语解释该方法、常见例题和考试技巧。
1. What Is an Implicit Function? | 什么是隐函数？
ich x and y are mixed together, for example x² + y² = 25, x³y + y² = 4, or xy + sin y = 1.
在 A-Level 数学中，你通常会遇到显函数，例如 y = 3x² - 5x + 2，其中 y 直接用 x 表示。隐函数是任何 x 与 y 混合在一起的方程，例如 x² + y² = 25、x³y + y² = 4 或 xy + sin y = 1。
The word 'implicit' means that y is not isolated. Even if it is possible to rearrange some equations, the result may be messy or involve multiple branches.
'隐式' 的意思是 y 没有被单独分离出来。即使有些方程可以整理，结果也可能很复杂或包含多个分支。
Implicit differentiation treats y as an unknown function of x, so we can differentiate directly without solving for y first.
隐式求导把 y 看作 x 的未知函数，因此我们可以直接求导，而不需要先解出 y。
2. Why We Need Implicit Differentiation | 为什么需要隐式求导
= 6xy has no simple explicit form, so ordinary differentiation cannot be applied directly.
有些隐式方程无法用标准函数整理成 y = f(x) 的形式。例如 x³ + y³ = 6xy 没有简单的显式形式，因此不能直接使用普通求导。
implicit form is often much shorter and avoids square roots, ± signs, and absolute values.
即使可以整理，对隐式形式求导通常也更简短，并且避免平方根、正负号和绝对值。
Edexcel exam questions therefore expect you to differentiate both sides with respect to x and apply the chain rule to y.
因此，Edexcel 考试题要求你对两边关于 x 求导，并对 y 使用链式法则。
3. The Core Method: Differentiate Both Sides | 核心方法：对方程两边求导
The fundamental idea is to differentiate every term with respect to x, remembering that y is a function of x. Unlike ordinary differentiation, the derivative of y is not 1 but dy/dx.
基本思想是对每一项关于 x 求导，同时记住 y 是 x 的函数。与普通求导不同，y 的导数不是 1，而是 dy/dx。
= 3x², but d/dx(y³) must be handled with the chain rule. Constants differentiate to zero.
例如，d/dx(x³) = 3x²，但 d/dx(y³) 必须用链式法则处理。常数求导为零。
After differentiation, collect all terms containing dy/dx on one side and factor it out before solving.
求导后，把所有包含 dy/dx 的项移到一边，并在求解前提取公因式。
4. Applying the Chain Rule to y | 对 y 使用链式法则
The chain rule states that d/dx[f(y)] = f'(y) × dy/dx. This is the single most important rule in implicit differentiation.
链式法则指出 d/dx[f(y)] = f'(y) × dy/dx。这是隐式求导中最重要的一条法则。
y results are: d/dx(y) = dy/dx, d/dx(y²) = 2y dy/dx, d/dx(sin y) = cos y dy/dx, and d/dx(eʸ) = eʸ dy/dx.
一些关键结果是：d/dx(y) = dy/dx，d/dx(y²) = 2y dy/dx，d/dx(sin y) = cos y dy/dx，以及 d/dx(eʸ) = eʸ dy/dx。
Notice that each derivative of a function of y is multiplied by dy/dx. This
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