# 合成関数の２次偏導関数の導出

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

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

もしくは，

$\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}$$+\frac{{\partial }^{2}z}{\partial {y}^{2}}{\left(\frac{dy}{dt}\right)}^{2}$$+\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

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

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

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

$=\left\{\frac{d}{dt}\left(\frac{\partial z}{\partial x}\right)\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{dx}{dt}+\frac{\partial z}{\partial x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{d}{dt}\left(\frac{dx}{dt}\right)\right\}$$+\left\{\frac{d}{dt}\left(\frac{\partial z}{\partial y}\right)\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{dy}{dt}+\frac{\partial z}{\partial y}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{d}{dt}\left(\frac{dy}{dt}\right)\right\}$

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

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

$\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}$$+\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial y}\right)\cdot \frac{dy}{dt}$

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

これらを代入して，

$\frac{{d}^{2}z}{d{t}^{2}}=\left\{\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}$$+\frac{\partial z}{\partial x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{d}{dt}\left(\frac{dx}{dt}\right)\right\}$$+\left\{\left(\frac{{\partial }^{2}z}{\partial x\partial y}\frac{dx}{dt}$$+\frac{{\partial }^{2}z}{\partial {y}^{2}}\frac{dy}{dt}\right)\frac{dy}{dt}$ $+\frac{\partial z}{\partial y}\cdot \frac{d}{dt}\left(\frac{dy}{dt}\right)\right\}$

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

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

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

$={f}_{xx}{\left(\frac{dx}{dt}\right)}^{2}+2{f}_{yx}\frac{dx}{dt}\frac{dy}{dt}$$+{f}_{yy}{\left(\frac{dy}{dt}\right)}^{2}$$+{f}_{x}\frac{{d}^{2}x}{d{t}^{2}}+{f}_{y}\frac{{d}^{2}y}{d{t}^{2}}$

${f}_{yx}={f}_{xy}$ より（ここを参照）

$={f}_{xx}{\left(\frac{dx}{dt}\right)}^{2}+2{f}_{xy}\frac{dx}{dt}\frac{dy}{dt}$$+{f}_{yy}{\left(\frac{dy}{dt}\right)}^{2}$$+{f}_{x}\frac{{d}^{2}x}{d{t}^{2}}+{f}_{y}\frac{{d}^{2}y}{d{t}^{2}}$

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