d/(dv)((p)(p+(9.3)/(v^2))(v-3.7))

解

\frac{d}{d v} ((p (v)) (p (v) + \frac{9.3}{v ^{2}}) (v - 3.7))

2 v p \frac{d p}{d v} - 7.4 p \frac{d p}{d v} + \frac{9.3 \frac{d p}{d v}}{v} + \frac{- 9.3 p - 34.41 \frac{d p}{d v}}{v ^{2}} + \frac{68.82 p}{v ^{3}} + p^{2}

解答ステップ

\frac{d}{d v} ((p) (p + \frac{9.3}{v ^{2}}) (v - 3.7))

積の法則を適用:

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

f = p, g = (p + \frac{9.3}{v ^{2}}) (v - 3.7)

= \frac{d p}{d v} (p + \frac{9.3}{v ^{2}}) (v - 3.7) + \frac{d}{d v} ((p + \frac{9.3}{v ^{2}}) (v - 3.7)) p

\frac{d}{d v} ((p + \frac{9.3}{v ^{2}}) (v - 3.7)) = v \frac{d p}{d v} - 3.7 \frac{d p}{d v} - \frac{9.3}{v ^{2}} + \frac{68.82}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{9.3}{v ^{2}}) (v - 3.7) + (v \frac{d p}{d v} - 3.7 \frac{d p}{d v} - \frac{9.3}{v ^{2}} + \frac{68.82}{v ^{3}} + p) p

簡素化

\frac{d p}{d v} (p + \frac{9.3}{v ^{2}}) (v - 3.7) + (v \frac{d p}{d v} - 3.7 \frac{d p}{d v} - \frac{9.3}{v ^{2}} + \frac{68.82}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 7.4 p \frac{d p}{d v} + \frac{9.3 \frac{d p}{d v}}{v} + \frac{- 9.3 p - 34.41 \frac{d p}{d v}}{v ^{2}} + \frac{68.82 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 7.4 p \frac{d p}{d v} + \frac{9.3 \frac{d p}{d v}}{v} + \frac{- 9.3 p - 34.41 \frac{d p}{d v}}{v ^{2}} + \frac{68.82 p}{v ^{3}} + p^{2}