What is (\partial)/(\partial z)(x^2y+yz^3+z^5x) ?

The answer to (\partial)/(\partial z)(x^2y+yz^3+z^5x) is 5xz^4+3yz^2

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(\partial)/(\partial z)(x^2y+yz^3+z^5x)

\frac{\partial}{\partial z} (x^{2} y + y z^{3} + z^{5} x)

5 x z^{4} + 3 y z^{2}

Solution steps

\frac{\partial}{\partial z} (x^{2} y + y z^{3} + z^{5} x)

Treat

x, y

as constants

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial z} (x^{2} y) + \frac{\partial}{\partial z} (y z^{3}) + \frac{\partial}{\partial z} (z^{5} x)

\frac{\partial}{\partial z} (x^{2} y) = 0

\frac{\partial}{\partial z} (y z^{3}) = 3 y z^{2}

\frac{\partial}{\partial z} (z^{5} x) = 5 x z^{4}

= 0 + 3 y z^{2} + 5 x z^{4}

Simplify

= 5 x z^{4} + 3 y z^{2}

What is (\partial)/(\partial z)(x^2y+yz^3+z^5x) ?
The answer to (\partial)/(\partial z)(x^2y+yz^3+z^5x) is 5xz^4+3yz^2