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

解

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

\frac{6 x x ^{\frac{5}{2}} - 2 x x ^{\frac{1}{2}} + 9 x x - x ^{3} + x - 3 x ^{\frac{3}{2}}}{2 x ^{\frac{3}{2}}}

解答ステップ

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

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

\frac{d}{d x} (x^{3} - x + 3 x^{\frac{3}{2}}) = 3 x^{2} - 1 + \frac{9 x ^{\frac{1}{2}}}{2}

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

= \frac{( 3 x ^{2} - 1 + \frac{9 x ^{\frac{1}{2}}}{2} ) x - \frac{1}{2 x ^{\frac{1}{2}}} ( x ^{3} - x + 3 x ^{\frac{3}{2}} )}{( x ) ^{2}}

簡素化

\frac{( 3 x ^{2} - 1 + \frac{9 x ^{\frac{1}{2}}}{2} ) x - \frac{1}{2 x ^{\frac{1}{2}}} ( x ^{3} - x + 3 x ^{\frac{3}{2}} )}{( x ) ^{2}} : \frac{6 x x ^{\frac{5}{2}} - 2 x x ^{\frac{1}{2}} + 9 x x - x ^{3} + x - 3 x ^{\frac{3}{2}}}{2 x ^{\frac{3}{2}}}

= \frac{6 x x ^{\frac{5}{2}} - 2 x x ^{\frac{1}{2}} + 9 x x - x ^{3} + x - 3 x ^{\frac{3}{2}}}{2 x ^{\frac{3}{2}}}