derivative f(x)=(2x^{3/2}-x^2+4)/x

解

導関数

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

\frac{- x ^{2} + x ^{\frac{3}{2}} - 4}{x ^{2}}

解答ステップ

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

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

\frac{d x}{d x} = 1

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

簡素化

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

= \frac{- x ^{2} + x ^{\frac{3}{2}} - 4}{x ^{2}}