Transformational View of Derivatives

Core Sentence / 核心句

A derivative describes how a function locally scales and orients tiny separations in its input space.

中文理解：把 function 看成 input space 到 output space 的 mapping 时，导数描述一个点附近的微小间距被拉伸、压缩或翻转多少。

f (a + h) \approx f (a) + f^{'} (a) h

|f'(a)| > 1  -> local expansion
0 < |f'(a)| < 1 -> local contraction
f'(a) < 0 -> orientation reversal
f'(a) = 0 -> first-order collapse

用于理解 local linearization、function iteration、fixed-point stability，以及 derivative 向 Jacobian 和更高维 calculus 的推广。

slope 与 local scale factor 是同一个 derivative 的两种 visualization。不要把 transformation view 误当作另一种 derivative，也不要把 local behavior 当作 global behavior。

Chapter 12 · Mapping Between Number Lines
- a derivative is the local scale and orientation factor of a number-line transformation
Chapter 12 · Zero and Negative Derivatives
- zero collapses first-order variation and a negative derivative reverses orientation
Chapter 12 · Complements Graphs
- this view is designed to generalize beyond ordinary scalar graphs