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

解

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

\frac{x ^{2} + 3}{2 x ^{\frac{5}{2}}}

解答ステップ

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

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

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

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

簡素化

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

= \frac{x ^{2} + 3}{2 x ^{\frac{5}{2}}}