derivative 3x^{-3/2}+6x^{-1/2}+x^3-4

解

導関数

3 x^{- \frac{3}{2}} + 6 x^{- \frac{1}{2}} + x^{3} - 4

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

解答ステップ

\frac{d}{d x} (3 x^{- \frac{3}{2}} + 6 x^{- \frac{1}{2}} + x^{3} - 4)

和/差の法則を適用:

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

= \frac{d}{d x} (3 x^{- \frac{3}{2}}) + \frac{d}{d x} (6 x^{- \frac{1}{2}}) + \frac{d}{d x} (x^{3}) - \frac{d}{d x} (4)

\frac{d}{d x} (3 x^{- \frac{3}{2}}) = - \frac{9}{2 x ^{\frac{5}{2}}}

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

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

\frac{d}{d x} (4) = 0

= - \frac{9}{2 x ^{\frac{5}{2}}} - \frac{3}{x ^{\frac{3}{2}}} + 3 x^{2} - 0

簡素化

- \frac{9}{2 x ^{\frac{5}{2}}} - \frac{3}{x ^{\frac{3}{2}}} + 3 x^{2} - 0 : \frac{- 9 - 6 x + 6 x ^{\frac{9}{2}}}{2 x ^{\frac{5}{2}}}

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