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

解

導関数

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

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

解答ステップ

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

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

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

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

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