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

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

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

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

e^{\frac{y}{2}} + \frac{e ^{\frac{y}{2}} x ^{2} + e ^{\frac{y}{2}} y}{2}

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial y} (x^{2} + y) = 1

\frac{\partial}{\partial y} (e^{\frac{y}{2}}) = e^{\frac{y}{2}} \frac{1}{2}

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

Simplify

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

= e^{\frac{y}{2}} + \frac{e ^{\frac{y}{2}} x ^{2} + e ^{\frac{y}{2}} y}{2}

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