derivative f(x)=((x-5))/((x+3)^4)

解

導関数

f (x) = \frac{( x - 5 )}{( x + 3 ) ^{4}}

\frac{- 3 x + 23}{( x + 3 ) ^{5}}

解答ステップ

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

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

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

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

簡素化

\frac{1 \cdot ( x + 3 ) ^{4} - 4 ( x + 3 ) ^{3} ( x - 5 )}{( ( x + 3 ) ^{4} ) ^{2}} : \frac{- 3 x + 23}{( x + 3 ) ^{5}}

= \frac{- 3 x + 23}{( x + 3 ) ^{5}}