(\partial ^2)/(\partial {h)(w)\partial w}(w/({h)(w)(w)^2})

解

\frac{\partial ^{3}}{\partial h ( w ) \partial w ^{2}} (\frac{w}{h ( w ) ( w ) ^{2}})

解答ステップ

\frac{\partial ^{3}}{\partial h ( w ) \partial w ^{2}} (\frac{w}{h ( w ) ( w ) ^{2}})

\frac{\partial}{\partial h ( w )} (\frac{w}{h ( w ) w ^{2}}) = - \frac{1}{w h ( w ) ^{2}}

= \frac{\partial ^{2}}{\partial w ^{2}} (- \frac{1}{w h ( w ) ^{2}})

\frac{\partial}{\partial w} (- \frac{1}{w h ( w ) ^{2}}) = \frac{h ( w ) + 2 w \frac{\partial}{\partial w} ( h ( w ) )}{w ^{2} h ( w ) ^{3}}

= \frac{\partial}{\partial w} (\frac{h ( w ) + 2 w \frac{\partial}{\partial w} ( h ( w ) )}{w ^{2} h ( w ) ^{3}})

\frac{\partial}{\partial w} (\frac{h ( w ) + 2 w \frac{\partial}{\partial w} ( h ( w ) )}{w ^{2} h ( w ) ^{3}}) = - \frac{2 ( 3 w ^{2} \frac{\partial}{\partial w} ( h ( w ) ) ^{2} - w ^{2} h ( w ) \frac{\partial ^{2}}{\partial w ^{2}} ( h ( w ) ) + 2 w h ( w ) \frac{\partial}{\partial w} ( h ( w ) ) + h ( w ) ^{2} )}{w ^{3} h ( w ) ^{4}}

= - \frac{2 ( 3 w ^{2} \frac{\partial}{\partial w} ( h ( w ) ) ^{2} - w ^{2} h ( w ) \frac{\partial ^{2}}{\partial w ^{2}} ( h ( w ) ) + 2 w h ( w ) \frac{\partial}{\partial w} ( h ( w ) ) + h ( w ) ^{2} )}{w ^{3} h ( w ) ^{4}}