derivative of (x^2(x^3-1)/(sqrt(x^2+1)))

解

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

\frac{4 x ^{6} + 5 x ^{4} - x ^{3} - 2 x}{( x ^{2} + 1 ) x ^{2} + 1}

解答ステップ

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

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

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

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

簡素化

\frac{( 5 x ^{4} - 2 x ) x ^{2} + 1 - \frac{x}{x ^{2} + 1} x ^{2} ( x ^{3} - 1 )}{( x ^{2} + 1 ) ^{2}} : \frac{4 x ^{6} + 5 x ^{4} - x ^{3} - 2 x}{( x ^{2} + 1 ) x ^{2} + 1}

= \frac{4 x ^{6} + 5 x ^{4} - x ^{3} - 2 x}{( x ^{2} + 1 ) x ^{2} + 1}