derivative f(x)=(x-7)/(9x^2e^x)

解

導関数

f (x) = \frac{x - 7}{9 x ^{2} e ^{x}}

\frac{- x ^{2} + 6 x + 14}{9 e ^{x} x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{x - 7}{9 x ^{2} e ^{x}})

定数を除去:

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

= \frac{1}{9} \frac{d}{d x} (\frac{x - 7}{x ^{2} e ^{x}})

商の法則を適用:

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

= \frac{1}{9} \cdot \frac{\frac{d}{d x} ( x - 7 ) x ^{2} e ^{x} - \frac{d}{d x} ( x ^{2} e ^{x} ) ( x - 7 )}{( x ^{2} e ^{x} ) ^{2}}

\frac{d}{d x} (x - 7) = 1

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

= \frac{1}{9} \cdot \frac{1 \cdot x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( x - 7 )}{( x ^{2} e ^{x} ) ^{2}}

簡素化

\frac{1}{9} \cdot \frac{1 \cdot x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( x - 7 )}{( x ^{2} e ^{x} ) ^{2}} : \frac{- x ^{2} + 6 x + 14}{9 e ^{x} x ^{3}}

= \frac{- x ^{2} + 6 x + 14}{9 e ^{x} x ^{3}}