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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

\frac{\partial}{\partial x} (2) = 0

= 2 e^{\frac{x}{2} + y} + e^{\frac{x + 2 y}{2}} x + 0

Simplify

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

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