derivative f(x)=(e^{x^2})/(x^3)

解

導関数

f (x) = \frac{e ^{x^{2}}}{x ^{3}}

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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