Chain Rule

Core Sentence / 核心句

The Chain Rule multiplies the rates at which a tiny change passes through successive function layers.

中文理解：输入变化经过多层函数时，每一层都按自己的局部倍率放大、缩小或反向这个变化；最终倍率是各层倍率的乘积。

\frac{d}{d x} g (h (x)) = g^{'} (h (x)) h^{'} (x)

Leibniz notation:

\frac{d g}{d x} = \frac{d g}{d h} \frac{d h}{d x}

Differentiate the outside at the unchanged inside, then multiply by the derivative of the inside.

中文：外导内不动，再乘内导。

最外层结构是 Function Composition，也就是一个函数的输出被送入另一个函数时使用。

微积分BC第3次正课 · Chain Rule
- local rates multiply as a tiny change passes through nested function layers
Chapter 7 · L’Hopital Example
- $d [sin (π x)] \approx π cos (π x) d x$ supplies the numerator’s local change
Chapter 6 · Differentiating the Constraint
- $x (t)^{2}$ differentiates to $2 x d x / d t$ because $x$ depends on time
Chapter 5 · Chain Rule Explains e^(kt)
- the inner rate $k$ multiplies the unchanged outer exponential
Chapter 4 · Chain Rule
- $d x \to d h \to df$ explains why successive local rates multiply