(\partial ^2)/(\partial {K)(b)\partial b}(a{K}(b)(b)^aL^b)

解

\frac{\partial ^{3}}{\partial K ( b ) \partial b ^{2}} (a K (b) (b)^{a} L^{b})

a (L^{b} b^{a} ln^{2} (L) + a^{2} L^{b} b^{a - 2} - a L^{b} b^{a - 2} + 2 a L^{b} b^{a - 1} ln (L))

解答ステップ

\frac{\partial ^{3}}{\partial K ( b ) \partial b ^{2}} (a K (b) (b)^{a} L^{b})

\frac{\partial}{\partial K ( b )} (a K (b) b^{a} L^{b}) = a L^{b} b^{a}

= \frac{\partial ^{2}}{\partial b ^{2}} (a L^{b} b^{a})

\frac{\partial}{\partial b} (a L^{b} b^{a}) = a (L^{b} ln (L) b^{a} + a b^{a - 1} L^{b})

= \frac{\partial}{\partial b} (a (L^{b} ln (L) b^{a} + a b^{a - 1} L^{b}))

\frac{\partial}{\partial b} (a (L^{b} ln (L) b^{a} + a b^{a - 1} L^{b})) = a (L^{b} b^{a} ln^{2} (L) + a^{2} L^{b} b^{a - 2} - a L^{b} b^{a - 2} + 2 a L^{b} b^{a - 1} ln (L))

= a (L^{b} b^{a} ln^{2} (L) + a^{2} L^{b} b^{a - 2} - a L^{b} b^{a - 2} + 2 a L^{b} b^{a - 1} ln (L))