What is (\partial)/(\partial z)(8ze^{xyz}) ?

The answer to (\partial)/(\partial z)(8ze^{xyz}) is 8(e^{zxy}+e^{zxy}zxy)

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(\partial)/(\partial z)(8ze^{xyz})

\frac{\partial}{\partial z} (8 z e^{x yz})

8 (e^{z x y} + e^{z x y} z x y)

Solution steps

\frac{\partial}{\partial z} (8 z e^{x yz})

Treat

x, y

as constants

Take the constant out:

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

= 8 \frac{\partial}{\partial z} (z e^{x yz})

Apply the Product Rule:

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

f = z, g = e^{x yz}

= 8 (\frac{\partial}{\partial z} (z) e^{x yz} + \frac{\partial}{\partial z} (e^{x yz}) z)

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

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

= 8 (1 \cdot e^{x yz} + e^{x yz} x yz)

Simplify

= 8 (e^{z x y} + e^{z x y} z x y)

What is (\partial)/(\partial z)(8ze^{xyz}) ?
The answer to (\partial)/(\partial z)(8ze^{xyz}) is 8(e^{zxy}+e^{zxy}zxy)