d/(dy)((q)(q^{0.3}+y^{0.3}))

Lösung

\frac{d}{d y} ((q (y)) (q (y)^{0.3} + y^{0.3}))

y^{0.3} \frac{d q}{d y} + 1.3 \frac{d q}{d y} q^{0.3} + \frac{0.3 q}{y ^{0.7}}

Schritte zur Lösung

\frac{d}{d y} ((q) (q^{0.3} + y^{0.3}))

Wende die Produktregel an:

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

f = q, g = q^{0.3} + y^{0.3}

= \frac{d q}{d y} (q^{0.3} + y^{0.3}) + \frac{d}{d y} (q^{0.3} + y^{0.3}) q

\frac{d}{d y} (q^{0.3} + y^{0.3}) = \frac{0.3 \frac{d q}{d y}}{q ^{0.7}} + \frac{0.3}{y ^{0.7}}

= \frac{d q}{d y} (q^{0.3} + y^{0.3}) + (\frac{0.3 \frac{d q}{d y}}{q ^{0.7}} + \frac{0.3}{y ^{0.7}}) q

Vereinfache

\frac{d q}{d y} (q^{0.3} + y^{0.3}) + (\frac{0.3 \frac{d q}{d y}}{q ^{0.7}} + \frac{0.3}{y ^{0.7}}) q : y^{0.3} \frac{d q}{d y} + 1.3 \frac{d q}{d y} q^{0.3} + \frac{0.3 q}{y ^{0.7}}

= y^{0.3} \frac{d q}{d y} + 1.3 \frac{d q}{d y} q^{0.3} + \frac{0.3 q}{y ^{0.7}}

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