Product Rule

Core Sentence / 核心句

Each term in the Product Rule represents one first-order contribution when both factors change.

中文理解：两个因子都随 $x$ 变化时，导数必须同时保留“第一个不动、第二个变化”和“第二个不动、第一个变化”两种一阶贡献。

Formula / 公式

$$\frac{d}{dx}[g(x)h(x)]=g(x)h^{\prime }(x)+h(x)g^{\prime }(x)$$

Differential form:

$$d(gh)\approx gdh+hdg$$

Visual Model / 图像模型

把 $g(x)h(x)$ 看成长方形面积：

• $gdh$ 是底部新增的薄条
• $hdg$ 是侧面新增的薄条
• $dgdh$ 是二阶角落小块，在 derivative limit 中不贡献

When To Use / 什么时候用

最外层 operation 是两个随 $x$ 变化的表达式相乘时使用。

Watch Out / 易错点

$$(gh{)}^{\prime }\ne g^{\prime }{h}^{\prime }$$

常数因子不需要 Product Rule 的完整形式，因为常数的导数为 $0$。

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 · Exponential Implicit Relation
  • the inner exponent xy contributes xy’+y before Chain Rule passes it through e
• 微积分BC第2次正课 · Product Rule
  • concrete, trigonometric, and abstract graph examples use the same two first-order contributions
• Chapter 6 · General Implicit Equation
  • differentiating $\sin (x)y^2$ requires both Product Rule and Chain Rule
• Chapter 4 · Product Rule Through Area
  • changing rectangle -> two first-order strips -> Product Rule

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
• 微积分BC第2次正课-极限应用与导数入门
• 04 - Visualizing the Chain Rule and Product Rule
• 06 - Implicit Differentiation, What’s Going On Here
• Implicit Differentiation
• Derivatives
• Tiny Nudges
• Derivative Formulas
• Chain Rule
• Function Composition