derivative of ae^v+b/v+c/(v^2)

解

\frac{d}{d x} (a e^{v (x)} + \frac{b}{v ( x )} + \frac{c}{v ( x ) ^{2}})

e^{v} \frac{d v}{d x} - \frac{b \frac{d v}{d x}}{v ^{2}} - \frac{2 c \frac{d v}{d x}}{v ^{3}}

解答ステップ

\frac{d}{d x} (a e^{v} + \frac{b}{v} + \frac{c}{v ^{2}})

和/差の法則を適用:

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

= \frac{d}{d x} (a e^{v}) + \frac{d}{d x} (\frac{b}{v}) + \frac{d}{d x} (\frac{c}{v ^{2}})

\frac{d}{d x} (a e^{v}) = e^{v} \frac{d v}{d x}

\frac{d}{d x} (\frac{b}{v}) = - \frac{b \frac{d v}{d x}}{v ^{2}}

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

= e^{v} \frac{d v}{d x} - \frac{b \frac{d v}{d x}}{v ^{2}} - \frac{2 c \frac{d v}{d x}}{v ^{3}}