d/(dv)((p)(p+(2.4)/(v^2))(v-8.3))

解答

\frac{d}{d v} ((p (v)) (p (v) + \frac{2.4}{v ^{2}}) (v - 8.3))

2 v p \frac{d p}{d v} - 16.6 p \frac{d p}{d v} + \frac{2.4 \frac{d p}{d v}}{v} + \frac{- 2.4 p - 19.92 \frac{d p}{d v}}{v ^{2}} + \frac{39.84 p}{v ^{3}} + p^{2}

求解步骤

\frac{d}{d v} ((p) (p + \frac{2.4}{v ^{2}}) (v - 8.3))

使用微分乘法定则:

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

f = p, g = (p + \frac{2.4}{v ^{2}}) (v - 8.3)

= \frac{d p}{d v} (p + \frac{2.4}{v ^{2}}) (v - 8.3) + \frac{d}{d v} ((p + \frac{2.4}{v ^{2}}) (v - 8.3)) p

\frac{d}{d v} ((p + \frac{2.4}{v ^{2}}) (v - 8.3)) = v \frac{d p}{d v} - 8.3 \frac{d p}{d v} - \frac{2.4}{v ^{2}} + \frac{39.84}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{2.4}{v ^{2}}) (v - 8.3) + (v \frac{d p}{d v} - 8.3 \frac{d p}{d v} - \frac{2.4}{v ^{2}} + \frac{39.84}{v ^{3}} + p) p

化简

\frac{d p}{d v} (p + \frac{2.4}{v ^{2}}) (v - 8.3) + (v \frac{d p}{d v} - 8.3 \frac{d p}{d v} - \frac{2.4}{v ^{2}} + \frac{39.84}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 16.6 p \frac{d p}{d v} + \frac{2.4 \frac{d p}{d v}}{v} + \frac{- 2.4 p - 19.92 \frac{d p}{d v}}{v ^{2}} + \frac{39.84 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 16.6 p \frac{d p}{d v} + \frac{2.4 \frac{d p}{d v}}{v} + \frac{- 2.4 p - 19.92 \frac{d p}{d v}}{v ^{2}} + \frac{39.84 p}{v ^{3}} + p^{2}

\frac{d ^{3}}{d t ^{3}} ((s) ((3 + 2 t)^{3}))

s im pl i f y (1 + \frac{2 i}{x})^{2}

e x p an d \frac{t ( t + 1 ) ( t - 1 )}{4 ( t + 3 )}

e x p an d (x - (\frac{3 + 2}{2})) (x - (\frac{3 - 2}{2}))

e x p an d (4 + 7) (6 + 5)