derivative of (2-x/((3-x)^4))

解

\frac{d}{d x} (\frac{2 - x}{( 3 - x ) ^{4}})

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

解答ステップ

\frac{d}{d x} (\frac{2 - x}{( 3 - x ) ^{4}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 2 - x ) ( 3 - x ) ^{4} - \frac{d}{d x} ( ( 3 - x ) ^{4} ) ( 2 - x )}{( ( 3 - x ) ^{4} ) ^{2}}

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

\frac{d}{d x} ((3 - x)^{4}) = - 4 (3 - x)^{3}

= \frac{( - 1 ) ( 3 - x ) ^{4} - ( - 4 ( 3 - x ) ^{3} ) ( 2 - x )}{( ( 3 - x ) ^{4} ) ^{2}}

簡素化

\frac{( - 1 ) ( 3 - x ) ^{4} - ( - 4 ( 3 - x ) ^{3} ) ( 2 - x )}{( ( 3 - x ) ^{4} ) ^{2}} : \frac{- 3 x + 5}{( 3 - x ) ^{5}}

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