derivative of (e^{2x}/(x^2))

解

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

\frac{2 e ^{2 x} ( x - 1 )}{x ^{3}}

解答ステップ

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

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

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

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

簡素化

\frac{e ^{2 x} \cdot 2 x ^{2} - 2 x e ^{2 x}}{( x ^{2} ) ^{2}} : \frac{2 e ^{2 x} ( x - 1 )}{x ^{3}}

= \frac{2 e ^{2 x} ( x - 1 )}{x ^{3}}