微分公式(1)
Author: 藍邦偉　老師

pf

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

$ =\lim_{ h \to 0 } \frac{(x+h)^n-x^n}{h}$

$ =\lim_{ h \to 0 }\frac{(C^n_0x^n+C^n_1x^{n-1}h+C^n_2x^{n-2}h²+...+C^n_{n-1}xh^{n-1}+C^n_nh^n)-x^n}{h} $

$ =\lim_{ h \to 0 }\frac{C^n_1x^{n-1}h+C^n_2x^{n-2}h²+...+C^n_{n-1}xh^{n-1}+C^n_nh^n}{h} $

$ =\lim_{ h \to 0 } \left( C^n_1x^{n-1}+C^n_2x^{n-2}h+...+C^n_{n-1}xh^{n-2}+C^n_nh^{n-1}  \right) $

=nx^{n-1}

$ 微分公式(1) $

n = 1 (slider from 1 to 4, step 1)
Q_1: "f(x)=x^n⇒f'(x)=nx^{n-1},n∈" + (LaTeX[If[n ≟ 1, "N", If[n ≟ 2, "Z", If[n ≟ 3, "Q", "R"]]]]) + ""
Step = 0 (slider from 0 to 6, step 1)