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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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