derivative of 2/3 x^{3/2}-1/2 x^{1/2}

解

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

\frac{- 1 + 4 x}{4 x ^{\frac{1}{2}}}

解答ステップ

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

簡素化

\frac{2}{3} x^{\frac{3}{2}} - \frac{1}{2} x^{\frac{1}{2}} : \frac{2 x ^{\frac{3}{2}}}{3} - \frac{x ^{\frac{1}{2}}}{2}

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

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= x^{\frac{1}{2}} - \frac{1}{4 x ^{\frac{1}{2}}}

簡素化

x^{\frac{1}{2}} - \frac{1}{4 x ^{\frac{1}{2}}} : \frac{- 1 + 4 x}{4 x ^{\frac{1}{2}}}

= \frac{- 1 + 4 x}{4 x ^{\frac{1}{2}}}