微分公式(3)
Author: 藍邦偉　老師

微分公式(3)

$ \frac{d}{dx} \left(f(x) \cdot g(x)   \right) $

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

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

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

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

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

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

$ \frac{d}{dx}\left(\frac{f(x)}{g(x)}   \right)  $

$ =\lim_{ h\to 0 }\frac{1}{h}\cdot \left [ \frac{f(x+h)}{g(x+h)}-\frac{f(x)}{g(x)} \right ] $

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

=\lim_{h \to 0 }\frac{f(x+h)g(x)-f(x)g(x)-f(x)g(x+h)+f(x)g(x)}{h}\cdot \lim_{ h \to 0 }\frac{1}{g(x)\cdot g(x+h)}

=\frac{1}{[g(x)]²}\cdot\lim_{ h \to 0 }\frac{(f(x+h)g(x)-f(x)g(x))-(f(x)g(x+h)-f(x)g(x))}{h}

$ =\frac{1}{[g(x)]²}\cdot \left(  g(x) \cdot\lim_{ h \to 0 }\frac{f(x+h)-f(x)}{h}-f(x)\cdot \lim_{ h \to 0 }\frac{g(x+h)-g(x)}{h} \right)  $

$ =\frac{f'(x)g(x)-g'(x)f(x)}{[g(x)]²} $

n = 1 (slider from 1 to 4, step 1)
F1F2: "$ \frac{d}{dx}\left(f(x)" + (LaTeX[If[n ≟ 1, "+", "- "]]) + "g(x)  \right)  $"
A = (-3.86, 4.84)
Step1: "$ =\lim_{ h \to 0 }\frac{\left( f(x+h)" + (LaTeX[If[n ≟ 1, "+", "-"]]) + "g(x+h)  \right) -\left(f(x)" + (LaTeX[If[n ≟ 1, "+", "-"]]) + "g(x)   \right) }{h} $"
Step2: "$ =\lim_{ h \to 0 }\frac{(f(x+h)-f(x))" + (LaTeX[If[n ≟ 1, "+", "-"]]) + "(g(x+h)-g(x))}{h} $"
Step3: "$ =\lim_{ h \to 0 }\frac{f(x+h)-f(x)}{h}" + (LaTeX[If[n ≟ 1, "+", "-"]]) + "\lim_{ h \to 0 }\frac{g(x+h)-g(x)}{h} $"
Step4: "$ =f'(x)" + (LaTeX[If[n ≟ 1, "+", "-"]]) + "g'(x) $"
Step = 0 (slider from 0 to 4, step 1)
step = 0 (slider from 0 to 6, step 1)
s = 0 (slider from 0 to 6, step 1)
an: "以下f(x),g(x)均為可微函數" + (LaTeX[If[n ≟ 4, "，g(x)不為0"]]) + ""