Implicit Differentiation

Core Sentence / 核心句

Implicit differentiation finds how interdependent variables must change together so that their defining relation remains true.

中文理解：隐式求导不是机械地补 $dy/dx$，而是在求为了维持原方程，$x$ 与 $y$ 的微小变化必须满足什么关系。

Formula / 公式

For:

$$x^2+y^2=R^2$$

differentiate with respect to $x$:

$$2x+2y\frac{dy}{dx}=0$$ $$\frac{dy}{dx}=-\frac{x}{y}$$

Memory Sentence / 记忆句

Differentiate both sides, treating $y$ as a function of $x$.

中文：两边同时对 $x$ 求导；遇到 $y$ 的函数时，用 chain rule 带出 $dy/dx$。

When To Use / 什么时候用

看到 Implicit Curve、mixed $x$ and $y$、constraint equation、inverse-function relation 或要求 implicit slope 时使用。

Watch Out / 易错点

• 不要漏掉 $dy/dx$。
• 不要在求导之前代入具体点。
• 原 equation 的两边必须继续变化相等。
• 求得的 slope expression 可能同时包含 $x$ 和 $y$，这是正常的。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第3次正课 · What Implicit Differentiation Is
  • differentiating a relation finds how x and y must change together to preserve it
• Chapter 6 · Circle Procedure
  • $2xdx+2ydy=0$ selects the tangent direction that preserves the circle relation
• Chapter 6 · General Example
  • product rule and chain rule work together in $\sin (x)y^2=x$

Connected Notes / 连接

• 微积分BC第3次正课-可导性链式法则与隐函数求导
• 06 - Implicit Differentiation, What’s Going On Here
• Implicit Curve
• Related Rates
• Chain Rule
• Product Rule
• Tangent Line
• Natural Logarithm