derivative of (x^{-1/2}+2x(7-x^{-1}))

解

\frac{d}{d x} ((x^{- \frac{1}{2}} + 2 x) (7 - x^{- 1}))

(- \frac{1}{2 x ^{\frac{3}{2}}} + 2) (7 - \frac{1}{x}) + \frac{1 + 2 x ^{\frac{3}{2}}}{x ^{\frac{5}{2}}}

解答ステップ

\frac{d}{d x} ((x^{- \frac{1}{2}} + 2 x) (7 - x^{- 1}))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{- \frac{1}{2}} + 2 x, g = 7 - x^{- 1}

= \frac{d}{d x} (x^{- \frac{1}{2}} + 2 x) (7 - x^{- 1}) + \frac{d}{d x} (7 - x^{- 1}) (x^{- \frac{1}{2}} + 2 x)

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

\frac{d}{d x} (7 - x^{- 1}) = \frac{1}{x ^{2}}

= (- \frac{1}{2 x ^{\frac{3}{2}}} + 2) (7 - x^{- 1}) + \frac{1}{x ^{2}} (x^{- \frac{1}{2}} + 2 x)

簡素化

(- \frac{1}{2 x ^{\frac{3}{2}}} + 2) (7 - x^{- 1}) + \frac{1}{x ^{2}} (x^{- \frac{1}{2}} + 2 x) : (- \frac{1}{2 x ^{\frac{3}{2}}} + 2) (7 - \frac{1}{x}) + \frac{1 + 2 x ^{\frac{3}{2}}}{x ^{\frac{5}{2}}}

= (- \frac{1}{2 x ^{\frac{3}{2}}} + 2) (7 - \frac{1}{x}) + \frac{1 + 2 x ^{\frac{3}{2}}}{x ^{\frac{5}{2}}}