derivative of (e^{5x}/(x^{20)})

解

\frac{d}{d x} (\frac{e ^{5 x}}{x ^{20}})

\frac{5 e ^{5 x} ( x - 4 )}{x ^{21}}

解答ステップ

\frac{d}{d x} (\frac{e ^{5 x}}{x ^{20}})

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 ^{5 x} ) x ^{20} - \frac{d}{d x} ( x ^{20} ) e ^{5 x}}{( x ^{20} ) ^{2}}

\frac{d}{d x} (e^{5 x}) = e^{5 x} \cdot 5

\frac{d}{d x} (x^{20}) = 20 x^{19}

= \frac{e ^{5 x} \cdot 5 x ^{20} - 20 x ^{19} e ^{5 x}}{( x ^{20} ) ^{2}}

簡素化

\frac{e ^{5 x} \cdot 5 x ^{20} - 20 x ^{19} e ^{5 x}}{( x ^{20} ) ^{2}} : \frac{5 e ^{5 x} ( x - 4 )}{x ^{21}}

= \frac{5 e ^{5 x} ( x - 4 )}{x ^{21}}