derivative of 2xe^x-1/(sqrt(x))

解

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

\frac{1 + 4 e ^{x} x ^{\frac{3}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{2 x ^{\frac{3}{2}}}

解答ステップ

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

和/差の法則を適用:

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

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

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

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

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

簡素化

2 (e^{x} + e^{x} x) - (- \frac{1}{2 x ^{\frac{3}{2}}}) : \frac{1 + 4 e ^{x} x ^{\frac{3}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{2 x ^{\frac{3}{2}}}

= \frac{1 + 4 e ^{x} x ^{\frac{3}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{2 x ^{\frac{3}{2}}}