d/(dv)((p)(p+(8.1)/(v^2))(v-1.1))

Lösung

\frac{d}{d v} ((p (v)) (p (v) + \frac{8.1}{v ^{2}}) (v - 1.1))

2 v p \frac{d p}{d v} - 2.2 p \frac{d p}{d v} + \frac{8.1 \frac{d p}{d v}}{v} + \frac{- 8.1 p - 8.91 \frac{d p}{d v}}{v ^{2}} + \frac{17.82 p}{v ^{3}} + p^{2}

Schritte zur Lösung

\frac{d}{d v} ((p) (p + \frac{8.1}{v ^{2}}) (v - 1.1))

Wende die Produktregel an:

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

f = p, g = (p + \frac{8.1}{v ^{2}}) (v - 1.1)

= \frac{d p}{d v} (p + \frac{8.1}{v ^{2}}) (v - 1.1) + \frac{d}{d v} ((p + \frac{8.1}{v ^{2}}) (v - 1.1)) p

\frac{d}{d v} ((p + \frac{8.1}{v ^{2}}) (v - 1.1)) = v \frac{d p}{d v} - 1.1 \frac{d p}{d v} - \frac{8.1}{v ^{2}} + \frac{17.82}{v ^{3}} + p

= \frac{d p}{d v} (p + \frac{8.1}{v ^{2}}) (v - 1.1) + (v \frac{d p}{d v} - 1.1 \frac{d p}{d v} - \frac{8.1}{v ^{2}} + \frac{17.82}{v ^{3}} + p) p

Vereinfache

\frac{d p}{d v} (p + \frac{8.1}{v ^{2}}) (v - 1.1) + (v \frac{d p}{d v} - 1.1 \frac{d p}{d v} - \frac{8.1}{v ^{2}} + \frac{17.82}{v ^{3}} + p) p : 2 v p \frac{d p}{d v} - 2.2 p \frac{d p}{d v} + \frac{8.1 \frac{d p}{d v}}{v} + \frac{- 8.1 p - 8.91 \frac{d p}{d v}}{v ^{2}} + \frac{17.82 p}{v ^{3}} + p^{2}

= 2 v p \frac{d p}{d v} - 2.2 p \frac{d p}{d v} + \frac{8.1 \frac{d p}{d v}}{v} + \frac{- 8.1 p - 8.91 \frac{d p}{d v}}{v ^{2}} + \frac{17.82 p}{v ^{3}} + p^{2}

e x p an d \frac{sin ( x ) + csc ( x ) + 2}{( csc ( x ) + 1 ) ( sin ( x ) + 1 )}

e x p an d \frac{x}{x ^{2}} - \frac{1}{x ^{2}}

e x p an d \frac{1 - x}{( x + 3 ) ^{2}}

e x p an d (3 x^{2} - 12) \cdot (x + 2 + 2)

e x p an d (cos (x) - 2) (cos (x) + 1)