d/(dv)((p)(p+(9.5)/(v^2))(v-9.8))

解答

\frac{d}{d v} ((p (v)) (p (v) + \frac{9.5}{v ^{2}}) (v - 9.8))

2 v p \frac{d p}{d v} - 19.6 p \frac{d p}{d v} + \frac{9.5 \frac{d p}{d v}}{v} + \frac{- 9.5 p - 93.1 \frac{d p}{d v}}{v ^{2}} + \frac{186.2 p}{v ^{3}} + p^{2}

求解步骤

\frac{d}{d v} ((p) (p + \frac{9.5}{v ^{2}}) (v - 9.8))

使用微分乘法定则:

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

f = p, g = (p + \frac{9.5}{v ^{2}}) (v - 9.8)

= \frac{d p}{d v} (p + \frac{9.5}{v ^{2}}) (v - 9.8) + \frac{d}{d v} ((p + \frac{9.5}{v ^{2}}) (v - 9.8)) p

\frac{d}{d v} ((p + \frac{9.5}{v ^{2}}) (v - 9.8)) = v \frac{d p}{d v} - 9.8 \frac{d p}{d v} - \frac{9.5}{v ^{2}} + \frac{186.2}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{9.5}{v ^{2}}) (v - 9.8) + (v \frac{d p}{d v} - 9.8 \frac{d p}{d v} - \frac{9.5}{v ^{2}} + \frac{186.2}{v ^{3}} + p) p

化简

\frac{d p}{d v} (p + \frac{9.5}{v ^{2}}) (v - 9.8) + (v \frac{d p}{d v} - 9.8 \frac{d p}{d v} - \frac{9.5}{v ^{2}} + \frac{186.2}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 19.6 p \frac{d p}{d v} + \frac{9.5 \frac{d p}{d v}}{v} + \frac{- 9.5 p - 93.1 \frac{d p}{d v}}{v ^{2}} + \frac{186.2 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 19.6 p \frac{d p}{d v} + \frac{9.5 \frac{d p}{d v}}{v} + \frac{- 9.5 p - 93.1 \frac{d p}{d v}}{v ^{2}} + \frac{186.2 p}{v ^{3}} + p^{2}

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