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

x \to 0 lim (\frac{x + 1}{x - 1})

x^{- \frac{4}{5}}

\frac{3}{x}

\int \frac{x + 1}{x - 1} d x

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

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