Higher-Order Derivatives

Core Sentence / 核心句

Higher-order derivatives are obtained by repeatedly differentiating, so each order measures how the previous order changes.

中文理解：高阶导数不是全新的运算，而是不断对前一阶导数再求导。

Formula / 公式或规则

$$f^{(n)}(x)=\frac{d^{n}f}{d{x}^n}$$

Interpretation / 直觉

$$f\to f^{\prime }\to f^{\prime \prime }\to f^{\prime \prime \prime }\to \cdots$$

• $f^{\prime }$：local change of $f$
• $f^{\prime \prime }$：local change of slope
• $f^{\prime \prime \prime }$：local change of the second derivative

When To Use / 什么时候用

看到 repeated differentiation、$f^{(n)}$、Taylor coefficients、position/velocity/acceleration chain，就想到 higher-order derivatives。

Watch Out / 易错点

$f^{(n)}$ 表示第 $n$ 阶导数，不是 $f$ 的 $n$ 次幂。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第4次正课 · Higher-Order Derivatives
  • repeated differentiation extends ordinary and implicit derivative calculations
• Differential Equations 01 · Derivatives Defined by State
• Chapter 10 · Repeated Derivative Operation
• Chapter 10 · Function Approximation

Connected Notes / 连接

• 微积分BC第4次正课-反函数高阶导数与运动应用
• One-Dimensional Motion
• 10 - Higher Order Derivatives
• Derivatives
• Second Derivative
• Taylor Series
• Acceleration
• Differential Equations
• 01 - Differential Equations, a Tourist’s Guide