## 2018年4月12日木曜日

### 数学 - 解析学 - 多変数の関数 - 陰関数(第2次導関数、グラディエント、内積、偏微分)

1. $\begin{array}{}g\text{'}\text{'}\left(x\right)\\ =-\frac{\frac{d}{\mathrm{dx}}{D}_{1}f·{D}_{2}f-{D}_{1}f·\frac{d}{\mathrm{dx}}{D}_{2}f}{{\left({D}_{2}f\right)}^{2}}\\ =-\frac{\left({D}_{1}^{2}f,{D}_{2}{D}_{1}f\right)·\left(1,g\text{'}\left(x\right)\right){D}_{2}f-{D}_{1}f\left({D}_{1}{D}_{2}f,{D}_{2}^{2}f\right)·\left(1,g\text{'}\left(x\right)\right)}{{\left({D}_{2}f\right)}^{2}}\\ =-\frac{\left({D}_{1}^{2}f+{D}_{2}{D}_{1}f·g\text{'}\left(x\right)\right){D}_{2}f-{D}_{1}f\left({D}_{1}{D}_{2}f+{D}_{2}^{2}f·g\text{'}\left(x\right)\right)}{{\left({D}_{2}f\right)}^{2}}\\ =-\frac{\left({D}_{1}^{2}f+\left({D}_{2}{D}_{1}f\right)\left(-\frac{{D}_{1}f}{{D}_{2}f}\right)\right){D}_{2}f-{D}_{1}f\left({D}_{1}{D}_{2}f+{D}_{2}^{2}f\left(-\frac{{D}_{1}f}{{D}_{2}f}\right)\right)}{{\left({D}_{2}f\right)}^{2}}\\ =\frac{-\left({D}_{1}^{2}f\right){\left({D}_{2}f\right)}^{2}+\left({D}_{2}{D}_{1}f\right)\left({D}_{1}f\right)\left({D}_{2}f\right)+\left({D}_{1}f\right)\left({D}_{1}{D}_{2}f\right)\left({D}_{2}f\right)-\left({D}_{2}^{2}f\right){\left({D}_{1}f\right)}^{2}}{{\left({D}_{2}f\right)}^{3}}\\ =\frac{-\left({D}_{1}^{2}f\right){\left({D}_{2}f\right)}^{2}+2\left({D}_{1}{D}_{2}f\right)\left({D}_{1}f\right)\left({D}_{2}f\right)-\left({D}_{2}^{2}f\right){\left({D}_{1}f\right)}^{2}}{{\left({D}_{2}f\right)}^{3}}\end{array}$