derivative f(x)=(x^3)/(x^2-1)

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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