derivative f(x)=(40x^5)/(e^x)

解

導関数

f (x) = \frac{40 x ^{5}}{e ^{x}}

\frac{40 x ^{4} ( 5 - x )}{e ^{x}}

解答ステップ

\frac{d}{d x} (\frac{40 x ^{5}}{e ^{x}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 40 \frac{d}{d x} (\frac{x ^{5}}{e ^{x}})

商の法則を適用:

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

= 40 \cdot \frac{\frac{d}{d x} ( x ^{5} ) e ^{x} - \frac{d}{d x} ( e ^{x} ) x ^{5}}{( e ^{x} ) ^{2}}

\frac{d}{d x} (x^{5}) = 5 x^{4}

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

= 40 \cdot \frac{5 x ^{4} e ^{x} - e ^{x} x ^{5}}{( e ^{x} ) ^{2}}

簡素化

40 \cdot \frac{5 x ^{4} e ^{x} - e ^{x} x ^{5}}{( e ^{x} ) ^{2}} : \frac{40 x ^{4} ( 5 - x )}{e ^{x}}

= \frac{40 x ^{4} ( 5 - x )}{e ^{x}}