(\partial)/(\partial y)(xyarcsin(yz))

Lösung

\frac{\partial}{\partial y} (x y arcsin (yz))

x (arcsin (yz) + \frac{yz}{1 - y ^{2} z ^{2}})

Schritte zur Lösung

\frac{\partial}{\partial y} (x y arcsin (yz))

Behandle

x, z

als Konstanten

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= x \frac{\partial}{\partial y} (y arcsin (yz))

Wende die Produktregel an:

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

f = y, g = arcsin (yz)

= x (\frac{\partial}{\partial y} (y) arcsin (yz) + \frac{\partial}{\partial y} (arcsin (yz)) y)

\frac{\partial}{\partial y} (y) = 1

\frac{\partial}{\partial y} (arcsin (yz)) = \frac{z}{1 - z ^{2} y ^{2}}

= x (1 \cdot arcsin (yz) + \frac{z}{1 - z ^{2} y ^{2}} y)

Vereinfache

x (1 \cdot arcsin (yz) + \frac{z}{1 - z ^{2} y ^{2}} y) : x (arcsin (yz) + \frac{yz}{1 - y ^{2} z ^{2}})

= x (arcsin (yz) + \frac{yz}{1 - y ^{2} z ^{2}})

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