derivative ((3x^2e^x-4))/(x^2)

解

導関数

\frac{( 3 x ^{2} e ^{x} - 4 )}{x ^{2}}

\frac{3 e ^{x} x ^{3} + 8}{x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{3 x ^{2} e ^{x} - 4}{x ^{2}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 3 x ^{2} e ^{x} - 4 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 3 x ^{2} e ^{x} - 4 )}{( x ^{2} ) ^{2}}

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

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

= \frac{3 ( e ^{x} x ^{2} + 2 e ^{x} x ) x ^{2} - 2 x ( 3 x ^{2} e ^{x} - 4 )}{( x ^{2} ) ^{2}}

簡素化

\frac{3 ( e ^{x} x ^{2} + 2 e ^{x} x ) x ^{2} - 2 x ( 3 x ^{2} e ^{x} - 4 )}{( x ^{2} ) ^{2}} : \frac{3 e ^{x} x ^{3} + 8}{x ^{3}}

= \frac{3 e ^{x} x ^{3} + 8}{x ^{3}}