What is (\partial)/(\partial x)((z)(xy+xz^3-2yz)) ?

The answer to (\partial)/(\partial x)((z)(xy+xz^3-2yz)) is z(z^3+y)

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(\partial)/(\partial x)((z)(xy+xz^3-2yz))

\frac{\partial}{\partial x} ((z) (x y + x z^{3} - 2 yz))

z (z^{3} + y)

Solution steps

\frac{\partial}{\partial x} ((z) (x y + x z^{3} - 2 yz))

Treat

z, y

as constants

Take the constant out:

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

= z \frac{\partial}{\partial x} (x y + x z^{3} - 2 yz)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= z (\frac{\partial}{\partial x} (x y) + \frac{\partial}{\partial x} (x z^{3}) - \frac{\partial}{\partial x} (2 yz))

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

\frac{\partial}{\partial x} (x z^{3}) = z^{3}

\frac{\partial}{\partial x} (2 yz) = 0

= z (y + z^{3} - 0)

Simplify

= z (z^{3} + y)

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What is (\partial)/(\partial x)((z)(xy+xz^3-2yz)) ?
The answer to (\partial)/(\partial x)((z)(xy+xz^3-2yz)) is z(z^3+y)