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

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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