## 2016年12月13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 - 曲線の傾き

1. $\begin{array}{l}\frac{2{\left(1+h\right)}^{2}-2}{\left(1+h\right)-1}=\frac{4h+2{h}^{2}}{h}=4+2h\\ 4\end{array}$

2. $\begin{array}{l}\frac{{\left(-1+h\right)}^{2}+1-\left(1+1\right)}{-1+h-\left(-1\right)}=\frac{-2h+{h}^{2}}{h}=-2+h\\ -2\end{array}$

3. $\begin{array}{l}\frac{2\left(2+h\right)-7-\left(2·2-7\right)}{h}=\frac{2h}{h}=2\\ 2\end{array}$

4. $\begin{array}{l}\frac{{\left(\frac{1}{2}+h\right)}^{3}-{\left(\frac{1}{2}\right)}^{3}}{h}\\ =\frac{{\left(\frac{1}{2}\right)}^{3}+3{\left(\frac{1}{2}\right)}^{2}h+3·\frac{1}{2}{h}^{2}+{h}^{3}-{\left(\frac{1}{2}\right)}^{3}}{h}\\ =\frac{3}{4}+\frac{3}{2}h+{h}^{2}\\ \frac{3}{4}\end{array}$

5. $\begin{array}{l}\frac{\frac{1}{2+h}-\frac{1}{2}}{h}=\frac{\frac{2-2-h}{2\left(2+h\right)}}{h}=\frac{-h}{2h\left(h+2\right)}=\frac{-1}{2h+4}\\ -\frac{1}{4}\end{array}$

6. $\begin{array}{l}\frac{{\left(-1+h\right)}^{2}+2\left(-1+h\right)-\left(1-2\right)}{h}\\ =\frac{1-2h+{h}^{2}-2+2h+1}{h}\\ =h\\ 0\end{array}$

7. $\begin{array}{l}\frac{{\left(2+h\right)}^{2}-{2}^{2}}{h}=\frac{4h+{h}^{2}}{h}=4+h\\ 4\end{array}$

8. $\begin{array}{l}\frac{{\left(3+h\right)}^{2}-{3}^{2}}{h}=\frac{6h+{h}^{2}}{h}=6+h\\ 6\end{array}$

9. $\begin{array}{l}\frac{{\left(1+h\right)}^{3}-{1}^{3}}{h}=\frac{3h+3{h}^{2}+{h}^{3}}{h}=3+3h+{h}^{2}\\ 3\end{array}$

10. $\begin{array}{l}\frac{{\left(2+h\right)}^{3}-{2}^{3}}{h}=\frac{12h+6{h}^{2}+{h}^{3}}{h}=12+6h+{h}^{2}\\ 12\end{array}$

11. $\begin{array}{l}\frac{2\left(2+h\right)+3-\left(2·2+3\right)}{h}=\frac{2h}{h}=2\\ 2\end{array}$

12. $\begin{array}{l}\frac{3\left(1+h\right)-5-\left(3-5\right)}{h}=\frac{3h}{h}=3\\ 3\end{array}$

13. $\begin{array}{l}\frac{a\left(x+h\right)+b-\left(ax+b\right)}{h}=\frac{ah}{h}=a\\ a\end{array}$