What is (\partial)/(\partial x)(xe^{sqrt(4xy)}) ?

The answer to (\partial)/(\partial x)(xe^{sqrt(4xy)}) is e^{2sqrt(xy)}sqrt(x)sqrt(y)+e^{2sqrt(xy)}

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(\partial)/(\partial x)(xe^{sqrt(4xy)})

\frac{\partial}{\partial x} (x e^{4 x y})

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

Solution steps

\frac{\partial}{\partial x} (x e^{4 x y})

Simplify

x e^{4 x y} : e^{2 x y} x

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial x)(xe^{sqrt(4xy)}) ?
The answer to (\partial)/(\partial x)(xe^{sqrt(4xy)}) is e^{2sqrt(xy)}sqrt(x)sqrt(y)+e^{2sqrt(xy)}