微分公式(4)
Author: 藍邦偉　老師

pf

令\Delta u=g(x+h)-g(x),g(x)=u

則：g(x+h)=u+ \Delta u，當h→0⇒\Delta u→0 

$∴[f(g(x)]'=\lim_{ h \to 0 }\frac{f[g(x+h)]-f[g(x)]}{h} $

$ =\lim_{ h \to 0 }\frac{f[g(x+h)]-f[g(x)]}{g(x+h)-g(x)}\cdot \frac{g(x+h)-g(x)}{h} $

$ =\lim_{ h \to 0 } \frac{f[g(x+h)]-f[g(x)]}{g(x+h)-g(x)}\cdot \lim_{ h \to 0 }\frac{g(x+h)-g(x)}{h}$

=\lim_{ \Delta u \to 0 }\frac{f(u+\Delta u)-f(u)}{\Delta u}\cdot g'(x)

$ 微分公式(4) Chain Rule $

$ =f'(u)\cdot g'(x) $

若g(x)在x可微，f(x)在g(x)可微，則f(g(x))在x可微. 而且y=f(g(x))，u=g(x)⇒\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}

$ =f'(g(x))\cdot g'(x) $

Step = 0 (slider from 0 to 8, step 1)