d/(dv)((p)((p+5/(v^2))(v-3.1)))

Lösung

\frac{d}{d v} ((p (v)) ((p (v) + \frac{5}{v ^{2}}) (v - 3.1)))

2 v p \frac{d p}{d v} - 6.2 p \frac{d p}{d v} + \frac{5 \frac{d p}{d v}}{v} + \frac{- 5 p - 15.5 \frac{d p}{d v}}{v ^{2}} + \frac{31 p}{v ^{3}} + p^{2}

Schritte zur Lösung

\frac{d}{d v} ((p) ((p + \frac{5}{v ^{2}}) (v - 3.1)))

Wende die Produktregel an:

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

f = p, g = (p + \frac{5}{v ^{2}}) (v - 3.1)

= \frac{d p}{d v} (p + \frac{5}{v ^{2}}) (v - 3.1) + \frac{d}{d v} ((p + \frac{5}{v ^{2}}) (v - 3.1)) p

\frac{d}{d v} ((p + \frac{5}{v ^{2}}) (v - 3.1)) = v \frac{d p}{d v} - 3.1 \frac{d p}{d v} - \frac{5}{v ^{2}} + \frac{31}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{5}{v ^{2}}) (v - 3.1) + (v \frac{d p}{d v} - 3.1 \frac{d p}{d v} - \frac{5}{v ^{2}} + \frac{31}{v ^{3}} + p) p

Vereinfache

\frac{d p}{d v} (p + \frac{5}{v ^{2}}) (v - 3.1) + (v \frac{d p}{d v} - 3.1 \frac{d p}{d v} - \frac{5}{v ^{2}} + \frac{31}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 6.2 p \frac{d p}{d v} + \frac{5 \frac{d p}{d v}}{v} + \frac{- 5 p - 15.5 \frac{d p}{d v}}{v ^{2}} + \frac{31 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 6.2 p \frac{d p}{d v} + \frac{5 \frac{d p}{d v}}{v} + \frac{- 5 p - 15.5 \frac{d p}{d v}}{v ^{2}} + \frac{31 p}{v ^{3}} + p^{2}

e x p an d (\frac{3 a + b}{4})^{2} - (\frac{b ^{2} + ba + a ^{2}}{2})

e x p an d l im x \Rightarrow 5 (x^{2} - x - 20 x^{2} - 7 x + 10)

e x p an d \frac{( x + h ) y ^{3} - ( x + h ) ^{3} y}{h}

s im pl i f y 8 (6 - 4)

e x p an d \frac{( x + 4 ) ^{4}}{4}