derivative of x^2-5x+1/(x^{3/2})

解

\frac{d}{d x} (x^{2} - 5 x + \frac{1}{x ^{\frac{3}{2}}})

\frac{- 3 + 4 x ^{\frac{7}{2}}}{2 x ^{\frac{5}{2}}} - 5

解答ステップ

\frac{d}{d x} (x^{2} - 5 x + \frac{1}{x ^{\frac{3}{2}}})

和/差の法則を適用:

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

= \frac{d}{d x} (x^{2}) - \frac{d}{d x} (5 x) + \frac{d}{d x} (\frac{1}{x ^{\frac{3}{2}}})

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

\frac{d}{d x} (5 x) = 5

\frac{d}{d x} (\frac{1}{x ^{\frac{3}{2}}}) = - \frac{3}{2 x ^{\frac{5}{2}}}

= 2 x - 5 - \frac{3}{2 x ^{\frac{5}{2}}}

分数を組み合わせる

- \frac{3}{2 x ^{\frac{5}{2}}} + 2 x : \frac{- 3 + 4 x ^{\frac{7}{2}}}{2 x ^{\frac{5}{2}}}

= \frac{- 3 + 4 x ^{\frac{7}{2}}}{2 x ^{\frac{5}{2}}} - 5