What is (\partial)/(\partial y)((x+y)e^{y^2}) ?

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

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(\partial)/(\partial y)((x+y)e^{y^2})

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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