合成関数の２次偏導関数の導出

$z = f (x, y)$ $z = f (x, y)$ で $x = φ (t), y = ψ (t)$ $x = φ (t), y = ψ (t)$ ならば，

$\frac{d^{2} z}{d t^{2}} = f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{x y} \frac{d x}{d t} \frac{d y}{d t}$ $\frac{d^{2} z}{d t^{2}} = f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{x y} \frac{d x}{d t} \frac{d y}{d t}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$

もしくは，

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

■導出

$\frac{d^{2} z}{d t^{2}} = \frac{d}{d t} (\frac{d z}{d t})$ $\frac{d^{2} z}{d t^{2}} = \frac{d}{d t} (\frac{d z}{d t})$

$= \frac{d}{d t} (\frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t})$ $= \frac{d}{d t} (\frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t})$

$= \frac{d}{d t} (\frac{\partial z}{\partial x} \frac{d x}{d t}) + \frac{d}{d t} (\frac{\partial z}{\partial y} \frac{d y}{d t})$ $= \frac{d}{d t} (\frac{\partial z}{\partial x} \frac{d x}{d t}) + \frac{d}{d t} (\frac{\partial z}{\partial y} \frac{d y}{d t})$

$= {\frac{d}{d t} (\frac{\partial z}{\partial x}) \cdot \frac{d x}{d t} + \frac{\partial z}{\partial x} \cdot \frac{d}{d t} (\frac{d x}{d t})}$ $= {\frac{d}{d t} (\frac{\partial z}{\partial x}) \cdot \frac{d x}{d t} + \frac{\partial z}{\partial x} \cdot \frac{d}{d t} (\frac{d x}{d t})}$ $+ {\frac{d}{d t} (\frac{\partial z}{\partial y}) \cdot \frac{d y}{d t} + \frac{\partial z}{\partial y} \cdot \frac{d}{d t} (\frac{d y}{d t})}$ $+ {\frac{d}{d t} (\frac{\partial z}{\partial y}) \cdot \frac{d y}{d t} + \frac{\partial z}{\partial y} \cdot \frac{d}{d t} (\frac{d y}{d t})}$

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

$\frac{d}{d t} (\frac{\partial z}{\partial x}) = \frac{\partial}{\partial x} (\frac{\partial z}{\partial x}) \cdot \frac{d x}{d t}$ $\frac{d}{d t} (\frac{\partial z}{\partial x}) = \frac{\partial}{\partial x} (\frac{\partial z}{\partial x}) \cdot \frac{d x}{d t}$ $+ \frac{\partial}{\partial y} (\frac{\partial z}{\partial x}) \cdot \frac{d y}{d t}$ $+ \frac{\partial}{\partial y} (\frac{\partial z}{\partial x}) \cdot \frac{d y}{d t}$

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

$\frac{d}{d t} (\frac{\partial z}{\partial y}) = \frac{\partial}{\partial x} (\frac{\partial z}{\partial y}) \cdot \frac{d x}{d t}$ $\frac{d}{d t} (\frac{\partial z}{\partial y}) = \frac{\partial}{\partial x} (\frac{\partial z}{\partial y}) \cdot \frac{d x}{d t}$ $+ \frac{\partial}{\partial y} (\frac{\partial z}{\partial y}) \cdot \frac{d y}{d t}$ $+ \frac{\partial}{\partial y} (\frac{\partial z}{\partial y}) \cdot \frac{d y}{d t}$

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

これらを代入して，

$\frac{d^{2} z}{d t^{2}} = {(\frac{\partial^{2} z}{\partial x^{2}} \frac{d x}{d t} + \frac{\partial^{2} z}{\partial y \partial x} \frac{d y}{d t}) \frac{d x}{d t}$ $\frac{d^{2} z}{d t^{2}} = {(\frac{\partial^{2} z}{\partial x^{2}} \frac{d x}{d t} + \frac{\partial^{2} z}{\partial y \partial x} \frac{d y}{d t}) \frac{d x}{d t}$ $+ \frac{\partial z}{\partial x} \cdot \frac{d}{d t} (\frac{d x}{d t})}$ $+ \frac{\partial z}{\partial x} \cdot \frac{d}{d t} (\frac{d x}{d t})}$ $+ {(\frac{\partial^{2} z}{\partial x \partial y} \frac{d x}{d t}$ $+ {(\frac{\partial^{2} z}{\partial x \partial y} \frac{d x}{d t}$ $+ \frac{\partial^{2} z}{\partial y^{2}} \frac{d y}{d t}) \frac{d y}{d t}$ $+ \frac{\partial^{2} z}{\partial y^{2}} \frac{d y}{d t}) \frac{d y}{d t}$ $+ \frac{\partial z}{\partial y} \cdot \frac{d}{d t} (\frac{d y}{d t})}$ $+ \frac{\partial z}{\partial y} \cdot \frac{d}{d t} (\frac{d y}{d t})}$

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

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

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

$= f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{y x} \frac{d x}{d t} \frac{d y}{d t}$ $= f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{y x} \frac{d x}{d t} \frac{d y}{d t}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$

$f_{y x} = f_{x y}$ $f_{y x} = f_{x y}$ より（ここを参照）

$= f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{x y} \frac{d x}{d t} \frac{d y}{d t}$ $= f_{x x} {(\frac{d x}{d t})}^{2} + 2 f_{x y} \frac{d x}{d t} \frac{d y}{d t}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{y y} {(\frac{d y}{d t})}^{2}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$ $+ f_{x} \frac{d^{2} x}{d t^{2}} + f_{y} \frac{d^{2} y}{d t^{2}}$

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最終更新日： 2023年1月21日