d/(dv)((p)(p+(4.1)/(v^2))(v-8.5))

解

\frac{d}{d v} ((p (v)) (p (v) + \frac{4.1}{v ^{2}}) (v - 8.5))

2 v p \frac{d p}{d v} - 17 p \frac{d p}{d v} + \frac{4.1 \frac{d p}{d v}}{v} + \frac{- 4.1 p - 34.85 \frac{d p}{d v}}{v ^{2}} + \frac{69.7 p}{v ^{3}} + p^{2}

解答ステップ

\frac{d}{d v} ((p) (p + \frac{4.1}{v ^{2}}) (v - 8.5))

積の法則を適用:

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

f = p, g = (p + \frac{4.1}{v ^{2}}) (v - 8.5)

= \frac{d p}{d v} (p + \frac{4.1}{v ^{2}}) (v - 8.5) + \frac{d}{d v} ((p + \frac{4.1}{v ^{2}}) (v - 8.5)) p

\frac{d}{d v} ((p + \frac{4.1}{v ^{2}}) (v - 8.5)) = v \frac{d p}{d v} - 8.5 \frac{d p}{d v} - \frac{4.1}{v ^{2}} + \frac{69.7}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{4.1}{v ^{2}}) (v - 8.5) + (v \frac{d p}{d v} - 8.5 \frac{d p}{d v} - \frac{4.1}{v ^{2}} + \frac{69.7}{v ^{3}} + p) p

簡素化

\frac{d p}{d v} (p + \frac{4.1}{v ^{2}}) (v - 8.5) + (v \frac{d p}{d v} - 8.5 \frac{d p}{d v} - \frac{4.1}{v ^{2}} + \frac{69.7}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 17 p \frac{d p}{d v} + \frac{4.1 \frac{d p}{d v}}{v} + \frac{- 4.1 p - 34.85 \frac{d p}{d v}}{v ^{2}} + \frac{69.7 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 17 p \frac{d p}{d v} + \frac{4.1 \frac{d p}{d v}}{v} + \frac{- 4.1 p - 34.85 \frac{d p}{d v}}{v ^{2}} + \frac{69.7 p}{v ^{3}} + p^{2}