derivative of ((x+3^2)/((x+2)^5(x+4)^7))

解

\frac{d}{d x} (\frac{( x + 3 ) ^{2}}{( x + 2 ) ^{5} ( x + 4 ) ^{7}})

\frac{2 ( x + 3 ) ( - 5 x ^{2} - 29 x - 43 )}{( x + 2 ) ^{6} ( x + 4 ) ^{8}}

解答ステップ

\frac{d}{d x} (\frac{( x + 3 ) ^{2}}{( x + 2 ) ^{5} ( x + 4 ) ^{7}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ( x + 3 ) ^{2} ) ( x + 2 ) ^{5} ( x + 4 ) ^{7} - \frac{d}{d x} ( ( x + 2 ) ^{5} ( x + 4 ) ^{7} ) ( x + 3 ) ^{2}}{( ( x + 2 ) ^{5} ( x + 4 ) ^{7} ) ^{2}}

\frac{d}{d x} ((x + 3)^{2}) = 2 (x + 3)

\frac{d}{d x} ((x + 2)^{5} (x + 4)^{7}) = 5 (x + 2)^{4} (x + 4)^{7} + 7 (x + 4)^{6} (x + 2)^{5}

= \frac{2 ( x + 3 ) ( x + 2 ) ^{5} ( x + 4 ) ^{7} - ( 5 ( x + 2 ) ^{4} ( x + 4 ) ^{7} + 7 ( x + 4 ) ^{6} ( x + 2 ) ^{5} ) ( x + 3 ) ^{2}}{( ( x + 2 ) ^{5} ( x + 4 ) ^{7} ) ^{2}}

簡素化

\frac{2 ( x + 3 ) ( x + 2 ) ^{5} ( x + 4 ) ^{7} - ( 5 ( x + 2 ) ^{4} ( x + 4 ) ^{7} + 7 ( x + 4 ) ^{6} ( x + 2 ) ^{5} ) ( x + 3 ) ^{2}}{( ( x + 2 ) ^{5} ( x + 4 ) ^{7} ) ^{2}} : \frac{2 ( x + 3 ) ( - 5 x ^{2} - 29 x - 43 )}{( x + 2 ) ^{6} ( x + 4 ) ^{8}}

= \frac{2 ( x + 3 ) ( - 5 x ^{2} - 29 x - 43 )}{( x + 2 ) ^{6} ( x + 4 ) ^{8}}