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

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

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(\partial)/(\partial x)((e^{x/2})(2xy))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\frac{x ^{2}}{x - 9}

x^{\frac{1}{3}}

(a - 7 x) (a + 7 x)

(x^{3} + 3 y) d x - x d y = 0

\frac{d y}{d x} - 4 y = - 4 y^{2}

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