derivative of (x^2/((x^2+5)^3))

解

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

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

解答ステップ

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

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

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

\frac{d}{d x} ((x^{2} + 5)^{3}) = 6 x (x^{2} + 5)^{2}

= \frac{2 x ( x ^{2} + 5 ) ^{3} - 6 x ( x ^{2} + 5 ) ^{2} x ^{2}}{( ( x ^{2} + 5 ) ^{3} ) ^{2}}

簡素化

\frac{2 x ( x ^{2} + 5 ) ^{3} - 6 x ( x ^{2} + 5 ) ^{2} x ^{2}}{( ( x ^{2} + 5 ) ^{3} ) ^{2}} : \frac{2 x ( - 2 x ^{2} + 5 )}{( x ^{2} + 5 ) ^{4}}

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