# 合成関数の2次偏導関数

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

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

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