What is (\partial)/(\partial y)(3xy^2e^{4yz}) ?

The answer to (\partial)/(\partial y)(3xy^2e^{4yz}) is 3x(2ye^{4yz}+e^{4yz}*4zy^2)

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(\partial)/(\partial y)(3xy^2e^{4yz})

\frac{\partial}{\partial y} (3 x y^{2} e^{4 yz})

3 x (2 y e^{4 yz} + e^{4 yz} \cdot 4 z y^{2})

Solution steps

\frac{\partial}{\partial y} (3 x y^{2} e^{4 yz})

Treat

x, z

as constants

Take the constant out:

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

= 3 x \frac{\partial}{\partial y} (y^{2} e^{4 yz})

Apply the Product Rule:

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

f = y^{2}, g = e^{4 yz}

= 3 x (\frac{\partial}{\partial y} (y^{2}) e^{4 yz} + \frac{\partial}{\partial y} (e^{4 yz}) y^{2})

\frac{\partial}{\partial y} (y^{2}) = 2 y

\frac{\partial}{\partial y} (e^{4 yz}) = e^{4 yz} \cdot 4 z

= 3 x (2 y e^{4 yz} + e^{4 yz} \cdot 4 z y^{2})

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What is (\partial)/(\partial y)(3xy^2e^{4yz}) ?
The answer to (\partial)/(\partial y)(3xy^2e^{4yz}) is 3x(2ye^{4yz}+e^{4yz}*4zy^2)