(\partial ^2)/(\partial {u)(v)\partial v}(e^{{u}(v)v})

解

\frac{\partial ^{3}}{\partial u ( v ) \partial v ^{2}} (e^{u (v) v})

解答ステップ

\frac{\partial ^{3}}{\partial u ( v ) \partial v ^{2}} (e^{u (v) v})

\frac{\partial}{\partial u ( v )} (e^{u (v) v}) = e^{u (v) v} v

= \frac{\partial ^{2}}{\partial v ^{2}} (e^{u (v) v} v)

\frac{\partial}{\partial v} (e^{u (v) v} v) = e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)}

= \frac{\partial}{\partial v} (e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)})

\frac{\partial}{\partial v} (e^{v u (v)} v (v \frac{\partial}{\partial v} (u (v)) + u (v)) + e^{v u (v)}) = e^{v u (v)} v^{3} \frac{\partial}{\partial v} (u (v))^{2} + 2 e^{v u (v)} v^{2} u (v) \frac{\partial}{\partial v} (u (v)) + e^{v u (v)} v^{2} \frac{\partial ^{2}}{\partial v ^{2}} (u (v)) + e^{v u (v)} v u (v)^{2} + 4 e^{v u (v)} v \frac{\partial}{\partial v} (u (v)) + 2 e^{v u (v)} u (v)

= e^{v u (v)} v^{3} \frac{\partial}{\partial v} (u (v))^{2} + 2 e^{v u (v)} v^{2} u (v) \frac{\partial}{\partial v} (u (v)) + e^{v u (v)} v^{2} \frac{\partial ^{2}}{\partial v ^{2}} (u (v)) + e^{v u (v)} v u (v)^{2} + 4 e^{v u (v)} v \frac{\partial}{\partial v} (u (v)) + 2 e^{v u (v)} u (v)