## 2017年1月10日火曜日

### 数学 - 解析学 - 微分と基本的な関数 - 微分係数、導関数 - 導関数(累乗、絶対値、左微分係数、右微分係数)

1. $\begin{array}{l}x\ge 0\\ f\left(x\right)={x}^{2}\\ \underset{h\to 0,h>0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{{h}^{2}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}h\\ =0\\ \\ x<0\\ f\left(x\right)=-{x}^{2}\\ \underset{h\to 0,h<0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{-{h}^{2}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}-h\\ =0\\ \\ f\text{'}\left(0\right)=0\end{array}$

2. $\begin{array}{l}x\ge 0\\ f\left(x\right)={x}^{3}\\ \underset{h\to 0,h>0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{{h}^{3}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}{h}^{2}\\ =0\\ \\ x<0\\ f\left(x\right)=-{x}^{3}\\ \underset{h\to 0,h<0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{-{h}^{3}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}-{h}^{2}\\ =0\\ \\ f\text{'}\left(0\right)=0\end{array}$

3. $\begin{array}{l}f\left(x\right)={x}^{n-1}|x|\\ x\ge 0\\ f\left(x\right)={x}^{n}\\ \underset{h\to 0,h>0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{{h}^{n}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}{h}^{n-1}\\ =0\\ \\ x<0\\ f\left(x\right)=-{x}^{n}\\ \underset{h\to 0,h<0}{\mathrm{lim}}\frac{f\left(h\right)-f\left(0\right)}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}\frac{-{h}^{n}-0}{h}\\ =\underset{h\to 0,h>0}{\mathrm{lim}}-{h}^{n-1}\\ =0\\ \\ f\text{'}\left(0\right)=0\end{array}$