(\partial ^3)/(\partial x\partial y\partial z)(e^{xyz^5})

解

\frac{\partial ^{3}}{\partial x \partial y \partial z} (e^{x y z^{5}})

5 e^{x y z^{5}} x^{2} y^{2} z^{14} + 15 e^{x y z^{5}} x y z^{9} + 5 e^{x y z^{5}} z^{4}

解答ステップ

\frac{\partial ^{3}}{\partial x \partial y \partial z} (e^{x y z^{5}})

\frac{\partial}{\partial x} (e^{x y z^{5}}) = e^{x y z^{5}} y z^{5}

= \frac{\partial ^{2}}{\partial y \partial z} (e^{x y z^{5}} y z^{5})

\frac{\partial}{\partial y} (e^{x y z^{5}} y z^{5}) = z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})

= \frac{\partial}{\partial z} (z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}}))

\frac{\partial}{\partial z} (z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})) = 5 e^{x y z^{5}} x^{2} y^{2} z^{14} + 15 e^{x y z^{5}} x y z^{9} + 5 e^{x y z^{5}} z^{4}

= 5 e^{x y z^{5}} x^{2} y^{2} z^{14} + 15 e^{x y z^{5}} x y z^{9} + 5 e^{x y z^{5}} z^{4}