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

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

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

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

e^{z x y} + e^{z x y} z x y

Solution steps

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

Treat

x, y

as constants

Apply the Product Rule:

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

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

= \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

= 1 \cdot e^{x yz} + e^{x yz} x yz

Simplify

= e^{z x y} + e^{z x y} z x y

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

\frac{\partial}{\partial y} ((x + y)^{3} - 12 x y)

\int x \cdot sin (2 x^{2}) d x

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

\frac{d y}{d x} = (\frac{1 - y ^{2}}{1 - x ^{2}})^{\frac{1}{2}}

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