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

解

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

\frac{1}{6 x ^{\frac{1}{2}}} - \frac{3 x ^{\frac{1}{2}}}{2}

解答ステップ

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

簡素化

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

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

和/差の法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{\frac{1}{2}}}{3}) = \frac{1}{6 x ^{\frac{1}{2}}}

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

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