What is (\partial)/(\partial y)(xyarcsin(yz)) ?

The answer to (\partial)/(\partial y)(xyarcsin(yz)) is x(arcsin(yz)+(yz)/(sqrt(1-y^2z^2)))

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

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

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

Solution steps

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

Treat

x, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

(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)

Simplify

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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What is (\partial)/(\partial y)(xyarcsin(yz)) ?
The answer to (\partial)/(\partial y)(xyarcsin(yz)) is x(arcsin(yz)+(yz)/(sqrt(1-y^2z^2)))