What is (\partial)/(\partial y)(e^{xyz^2}yz^2) ?

The answer to (\partial)/(\partial y)(e^{xyz^2}yz^2) is z^2(e^{xyz^2}xyz^2+e^{xyz^2})

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(\partial)/(\partial y)(e^{xyz^2}yz^2)

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

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

Solution steps

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

Treat

x, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

f = e^{x y z^{2}}, g = y

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

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

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

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

Simplify

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

\frac{d}{d x} ((2 x + 1)^{4} + 8 x (2 x + 1)^{3})

\int \frac{28 e ^{4 x}}{6 + e ^{4 x}} d x

t e^{10 t}

x \to 0 lim (\frac{( arctan ( 4 x ) )}{ln ( x )})

\frac{\partial}{\partial x} (ln (4 x + 5 y))

What is (\partial)/(\partial y)(e^{xyz^2}yz^2) ?
The answer to (\partial)/(\partial y)(e^{xyz^2}yz^2) is z^2(e^{xyz^2}xyz^2+e^{xyz^2})