What is (\partial)/(\partial x)(2xe^{sqrt(xy)}) ?

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

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(\partial)/(\partial x)(2xe^{sqrt(xy)})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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