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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

= \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}}

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

Simplify

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

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

\frac{\partial}{\partial x} (2 y arctan (x))

y^{^{''}} + 10 y^{^{'}} + 61 y = 0

\frac{d}{d x} (\frac{x + 2 x ^{2}}{1 + x ^{2}})

x \to 4 - lim (\frac{3}{x - 2})

\int (x + \frac{1}{5 x}) d x

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