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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= \frac{1}{y} \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}

= \frac{1}{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}) = e^{x y} y

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

Simplify

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

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

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