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

The answer to (\partial)/(\partial y)(xye^{x/y}) is x(e^{x/y}-(e^{x/y}x)/y)

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(\partial)/(\partial y)(xye^{x/y})

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= x (e^{\frac{x}{y}} - \frac{e ^{\frac{x}{y}} x}{y})

\frac{d}{d x} ((6 x)^{5 x})

(x - y arctan (\frac{y}{x})) d x + x arctan (\frac{y}{x}) d y = 0

h (x) = \frac{2}{x ^{5}} + \frac{1}{x ^{7}}

\int_{- 1}^{1} x (x^{2} + 1)^{3} d x

x^{2} y^{^{''}} + x y^{^{'}} - 4 y = x^{2} + x

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