合成関数の２次偏導関数の導出

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

${\displaystyle \frac{d^{2}z}{d{t}^2}=f_{xx}{\left(\frac{dx}{dt}\right)}^2+2f_{xy}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +f_{yy}{\left(\frac{dy}{dt}\right)}^2}$${\displaystyle +f_x\frac{d^{2}x}{d{t}^2}+f_y\frac{d^{2}y}{d{t}^2}}$

もしくは，

${\displaystyle \frac{d^{2}z}{d{t}^2}=\frac{{\partial }^{2}z}{\partial x^2}{\left(\frac{dx}{dt}\right)}^2+2\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial y^2}{\left(\frac{dy}{dt}\right)}^2}$${\displaystyle +\frac{\partial z}{\partial x}\frac{d^{2}x}{d{t}^2}+\frac{\partial z}{\partial y}\frac{d^{2}y}{d{t}^2}}$

■導出

${\displaystyle \frac{d^{2}z}{d{t}^2}=\frac{d}{dt}\left(\frac{dz}{dt}\right)}$

${\displaystyle =\frac{d}{dt}\left(\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}\right)}$

${\displaystyle =\frac{d}{dt}\left(\frac{\partial z}{\partial x}\frac{dx}{dt}\right)+\frac{d}{dt}\left(\frac{\partial z}{\partial y}\frac{dy}{dt}\right)}$

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 ${\displaystyle t}$ で微分すると，

${\displaystyle \frac{d}{dt}\left(\frac{\partial z}{\partial x}\right)=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{dx}{dt}}$${\displaystyle +\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial x}\right)\cdot \frac{dy}{dt}}$

${\displaystyle =\frac{{\partial }^{2}z}{\partial x^2}\frac{dx}{dt}+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{dy}{dt}}$

${\displaystyle \frac{d}{dt}\left(\frac{\partial z}{\partial y}\right)=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{dx}{dt}}$${\displaystyle +\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{dy}{dt}}$

${\displaystyle =\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}+\frac{{\partial }^{2}z}{\partial y^2}\frac{dy}{dt}}$

これらを代入して，

${\displaystyle \frac{d^{2}z}{d{t}^2}=\{\left(\frac{{\partial }^{2}z}{\partial x^2}\frac{dx}{dt}+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{dy}{dt}\right)\frac{dx}{dt}}$${\displaystyle +\frac{\partial z}{\partial x}·\frac{d}{dt}\left(\frac{dx}{dt}\right)\}}$${\displaystyle +\{(\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial y^2}\frac{dy}{dt})\frac{dy}{dt}}$ ${\displaystyle +\frac{\partial z}{\partial y}\cdot \frac{d}{dt}\left(\frac{dy}{dt}\right)\}}$

${\displaystyle =\frac{{\partial }^{2}z}{\partial x^2}{\left(\frac{dx}{dt}\right)}^2+\frac{{\partial }^{2}z}{\partial y\partial x}\frac{dy}{dt}\frac{dx}{dt}}$${\displaystyle +\frac{\partial z}{\partial x}\frac{d^{2}x}{d{t}^2}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial y^2}{\left(\frac{dy}{dt}\right)}^2+\frac{\partial z}{\partial y}\frac{d^{2}y}{d{t}^2}}$

${\displaystyle =\frac{{\partial }^{2}z}{\partial x^2}{\left(\frac{dx}{dt}\right)}^2+\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +\frac{\partial z}{\partial x}\frac{d^{2}x}{d{t}^2}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial y^2}{\left(\frac{dy}{dt}\right)}^2+\frac{\partial z}{\partial y}\frac{d^{2}y}{d{t}^2}}$

${\displaystyle =\frac{{\partial }^{2}z}{\partial x^2}{\left(\frac{dx}{dt}\right)}^2+2\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +\frac{{\partial }^{2}z}{\partial y^2}{\left(\frac{dy}{dt}\right)}^2}$${\displaystyle +\frac{\partial z}{\partial x}\frac{d^{2}x}{d{t}^2}+\frac{\partial z}{\partial y}\frac{d^{2}y}{d{t}^2}}$

${\displaystyle =f_{xx}{\left(\frac{dx}{dt}\right)}^2+2f_{yx}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +f_{yy}{\left(\frac{dy}{dt}\right)}^2}$${\displaystyle +f_x\frac{d^{2}x}{d{t}^2}+f_y\frac{d^{2}y}{d{t}^2}}$

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

${\displaystyle =f_{xx}{\left(\frac{dx}{dt}\right)}^2+2f_{xy}\frac{dx}{dt}\frac{dy}{dt}}$${\displaystyle +f_{yy}{\left(\frac{dy}{dt}\right)}^2}$${\displaystyle +f_x\frac{d^{2}x}{d{t}^2}+f_y\frac{d^{2}y}{d{t}^2}}$

　

ホーム>>カテゴリー別分類>>微分>>偏微分>>合成関数の2次偏導関数>>合成関数の2次偏導関数の導出

学生スタッフ作成
最終更新日： 2023年1月21日