derivative of (3-1/x /(x+5))

解

\frac{d}{d x} (\frac{3 - \frac{1}{x}}{x + 5})

\frac{- 3 x ^{2} + 2 x + 5}{x ^{2} ( x + 5 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{3 - \frac{1}{x}}{x + 5})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 3 - \frac{1}{x} ) ( x + 5 ) - \frac{d}{d x} ( x + 5 ) ( 3 - \frac{1}{x} )}{( x + 5 ) ^{2}}

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

\frac{d}{d x} (x + 5) = 1

= \frac{\frac{1}{x ^{2}} ( x + 5 ) - 1 \cdot ( 3 - \frac{1}{x} )}{( x + 5 ) ^{2}}

簡素化

\frac{\frac{1}{x ^{2}} ( x + 5 ) - 1 \cdot ( 3 - \frac{1}{x} )}{( x + 5 ) ^{2}} : \frac{- 3 x ^{2} + 2 x + 5}{x ^{2} ( x + 5 ) ^{2}}

= \frac{- 3 x ^{2} + 2 x + 5}{x ^{2} ( x + 5 ) ^{2}}