# 合成関数の2次偏導関数の導出

$z=f\left(x,y\right)$$x=\phi \left(u,v\right),y=\psi \left(u,v\right)$ ならば

$\frac{{\partial }^{2}z}{\partial u\partial v}=\frac{{\partial }^{2}z}{\partial v\partial u}={f}_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}$$+{f}_{xy}\left(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}$$+\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right)$$+{f}_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}$$+{f}_{x}\frac{{\partial }^{2}x}{\partial u\partial v}+{f}_{y}\frac{{\partial }^{2}y}{\partial u\partial v}$

もしくは，

$\frac{{\partial }^{2}z}{\partial u\partial v}=\frac{{\partial }^{2}z}{\partial v\partial u}=\frac{{\partial }^{2}z}{\partial {x}^{2}}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}$$+\frac{{\partial }^{2}z}{\partial x\partial y}\left(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}$$+\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right)$$+\frac{{\partial }^{2}z}{{\partial }^{2}y}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}$$+\frac{\partial z}{\partial x}\frac{{\partial }^{2}x}{\partial u\partial v}+\frac{\partial z}{\partial y}\frac{{\partial }^{2}y}{\partial u\partial v}$

## ■導出

$\frac{{\partial }^{2}z}{\partial u\partial v}=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial v}\right)$

$=\frac{\partial }{\partial u}\left({f}_{x}\frac{\partial x}{\partial v}+{f}_{y}\frac{\partial y}{\partial v}\right)$

$=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}\right)$

$=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial v}\right)+\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial y}\frac{\partial y}{\partial v}\right)$

$=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{\partial x}{\partial v}+\frac{\partial z}{\partial x}\cdot \frac{\partial }{\partial u}\left(\frac{\partial x}{\partial v}\right)$$+\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{\partial y}{\partial v}$$+\frac{\partial z}{\partial y}\cdot \frac{\partial }{\partial u}\left(\frac{\partial y}{\partial v}\right)$

$=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{\partial x}{\partial v}+\frac{\partial z}{\partial x}\cdot \frac{{\partial }^{2}x}{\partial u\partial v}$$+\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{\partial y}{\partial v}$$+\frac{\partial z}{\partial y}\cdot \frac{{\partial }^{2}y}{\partial u\partial v}$

$z$合成関数であるので， $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial z}{\partial x}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial z}{\partial y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ も共に合成関数となる．これらを $\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\text{\hspace{0.17em}}\text{\hspace{0.17em}}$偏微分すると，合成関数の偏微分より，

$\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\right)=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{\partial x}{\partial u}$$+\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{\partial y}{\partial u}$

$=\frac{{\partial }^{2}z}{\partial {x}^{2}}\frac{\partial x}{\partial u}+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{\partial y}{\partial u}$

$\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial y}\right)=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{\partial x}{\partial u}$$+\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{\partial y}{\partial u}$

$=\frac{{\partial }^{2}z}{\partial x\partial y}\frac{\partial x}{\partial u}+\frac{{\partial }^{2}z}{\partial {y}^{2}}\frac{\partial y}{\partial u}$

これらを代入して，

$\frac{{\partial }^{2}z}{\partial u\partial v}=\left\{\left(\frac{{\partial }^{2}z}{\partial {x}^{2}}\frac{\partial x}{\partial u}+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{\partial y}{\partial u}\right)\frac{\partial x}{\partial u}$$+\frac{\partial z}{\partial x}\frac{{\partial }^{2}x}{\partial u\partial v}\right\}+\left\{\left(\frac{{\partial }^{2}z}{\partial x\partial y}\frac{\partial x}{\partial u}$$+\frac{{\partial }^{2}z}{\partial {y}^{2}}\frac{\partial y}{\partial u}\right)\frac{\partial y}{\partial u}+\frac{\partial z}{\partial y}\frac{{\partial }^{2}y}{\partial u\partial v}\right\}$

$=\left\{\frac{{\partial }^{2}z}{\partial {x}^{2}}\frac{\partial x}{\partial u}\cdot \frac{\partial x}{\partial v}+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{\partial y}{\partial u}\cdot \frac{\partial x}{\partial v}$$+\frac{\partial z}{\partial x}\cdot \frac{{\partial }^{2}x}{\partial u\partial v}\right\}$$+\left\{\frac{{\partial }^{2}z}{\partial x\partial y}\frac{\partial x}{\partial u}\cdot \frac{\partial y}{\partial v}$$+\frac{{\partial }^{2}z}{\partial {y}^{2}}\frac{\partial y}{\partial u}\cdot \frac{\partial y}{\partial v}+\frac{\partial z}{\partial y}\cdot \frac{{\partial }^{2}y}{\partial u\partial v}\right\}$

$={f}_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+{f}_{xy}\frac{\partial y}{\partial u}\frac{\partial x}{\partial v}$$+{f}_{x}\frac{{\partial }^{2}x}{\partial u\partial v}$$+{f}_{yx}\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}+{f}_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}$$+{f}_{y}\frac{{\partial }^{2}y}{\partial u\partial v}$

$={f}_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+{f}_{xy}\left(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right)$$+{f}_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}$$+{f}_{x}\frac{{\partial }^{2}x}{\partial u\partial v}+{f}_{y}\frac{{\partial }^{2}y}{\partial u\partial v}$

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