What is (\partial)/(\partial x)(xy^2e^{-xz}) ?

The answer to (\partial)/(\partial x)(xy^2e^{-xz}) is y^2(e^{-xz}-e^{-xz}xz)

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(\partial)/(\partial x)(xy^2e^{-xz})

\frac{\partial}{\partial x} (x y^{2} e^{- x z})

y^{2} (e^{- x z} - e^{- x z} x z)

Solution steps

\frac{\partial}{\partial x} (x y^{2} e^{- x z})

Treat

y, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

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

= y^{2} (\frac{\partial}{\partial x} (x) e^{- x z} + \frac{\partial}{\partial x} (e^{- x z}) x)

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

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

= y^{2} (1 \cdot e^{- x z} + (- z e^{- z x}) x)

Simplify

= y^{2} (e^{- x z} - e^{- x z} x z)

What is (\partial)/(\partial x)(xy^2e^{-xz}) ?
The answer to (\partial)/(\partial x)(xy^2e^{-xz}) is y^2(e^{-xz}-e^{-xz}xz)