合成関数の2次偏導関数の導出

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

${\displaystyle \frac{{\partial }^{2}z}{\partial u^2}=f_{xx}{\left(\frac{\partial x}{\partial u}\right)}^2+2f_{xy}\frac{\partial x}{\partial u}\frac{\partial y}{\partial u}}$${\displaystyle +f_{yy}{\left(\frac{\partial y}{\partial u}\right)}^2}$${\displaystyle +f_x\frac{{\partial }^{2}x}{\partial u^2}+f_y\frac{{\partial }^{2}y}{\partial u^2}}$

もしくは，

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

■導出

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

${\displaystyle =\frac{\partial }{\partial u}\left(f_x\frac{\partial x}{\partial u}+f_y\frac{\partial y}{\partial u}\right)}$

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

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

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

条件から， ${\displaystyle \frac{\partial z}{\partial x},\frac{\partial z}{\partial y}}$ は共に合成関数であるから，これらを ${\displaystyle u}$ で偏微分すると，

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

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

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

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

これらを代入して，

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

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

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

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

${\displaystyle f_{yx}=f_{xy}}$ より（ここを参照）

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最終更新日： 2023年1月21日