Power Rule

Core Sentence / 核心句

The Power Rule differentiates powers by bringing the exponent down and reducing it by one.

中文理解：幂函数求导时，指数先乘到前面，然后指数减 1。

Formula / 公式或规则

$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$

When To Use / 什么时候用

看到 polynomial derivative、Taylor coefficients、physics rate/change calculation，经常先用 power rule。

Watch Out / 易错点

不要把 power rule 用到不是 $x^n$ 的结构上；复合函数还要 chain rule。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第2次正课 · Basic Derivative Rules
  • the definition example for $x^2$ is compressed into the general Power Rule
• 03 - Derivative Formulas through Geometry · Knowledge Point 5: Power Rule Is Geometry, Not Exponent Magic
  • Knowledge Point 5: Power Rule Is Geometry, Not Exponent Magic
• 03 - Derivative Formulas through Geometry · 3. The Power Rule comes from the first-order terms in $(x+dx)^n$.
  • 3. The Power Rule comes from the first-order terms in $(x+dx)^n$.
• 03 - Derivative Formulas through Geometry · Knowledge Point 6: Why $\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$
  • 用 power rule 也能得到：
• AP电磁学第1次正课-电荷库仑定律与电场 · Derivatives and Integrals / 微积分工具
  • 这和 03 - Derivative Formulas through Geometry 直接相连：那篇解释了 Power Rule 和 Trigonometric Derivatives 的几何直觉，这里开始把它们作为物理工具使用。
• 03 - Derivative Formulas through Geometry · 4. Deep Phrases / 深句对应表
  • | the exponent hops down to the front | 这是 power rule 的计算口诀，不是原因。 |
• 03 - Derivative Formulas through Geometry · 7. Confusion / Questions
  • 2. Power Rule 里的 $n$ 为什么是“有 $n$ 个地方可以选 $dx$”？
• 03 - Derivative Formulas through Geometry · 2. Class Flow / 视频脉络
  • 4. 推广到 Power Rule：

Connected Notes / 连接

• 微积分BC第2次正课-极限应用与导数入门
• 03 - Derivative Formulas through Geometry
• Derivatives
• Taylor Series
• Trigonometric Derivatives