What is (\partial)/(\partial z)(-x^3yze^z) ?

The answer to (\partial)/(\partial z)(-x^3yze^z) is -x^3y(e^z+e^zz)

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(\partial)/(\partial z)(-x^3yze^z)

\frac{\partial}{\partial z} (- x^{3} yz e^{z})

- x^{3} y (e^{z} + e^{z} z)

Solution steps

\frac{\partial}{\partial z} (- x^{3} yz e^{z})

Treat

x, y

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

f = z, g = e^{z}

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

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

\frac{\partial}{\partial z} (e^{z}) = e^{z}

= - x^{3} y (1 \cdot e^{z} + e^{z} z)

Simplify

= - x^{3} y (e^{z} + e^{z} z)

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What is (\partial)/(\partial z)(-x^3yze^z) ?
The answer to (\partial)/(\partial z)(-x^3yze^z) is -x^3y(e^z+e^zz)