What is (\partial)/(\partial y)(e^{xyz^8}yz^8) ?

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

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(\partial)/(\partial y)(e^{xyz^8}yz^8)

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

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

Solution steps

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

Treat

x, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

f = e^{x y z^{8}}, g = y

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

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

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

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

Simplify

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

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