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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

= \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}

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

Simplify

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

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

y = x^{2} + 5 x + 3, at x = 1

\frac{d}{d x} (3 (5 - 7 x)^{5})

\frac{\partial}{\partial x} (x^{\frac{3}{4}} y^{- 4})

\int x sin^{2} (x) d x

\int (x^{2} - 1) e^{2 x} d x

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