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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{\partial}{\partial y} (x) = 0

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

= 0 + e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

Simplify

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

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

\frac{\partial}{\partial x} (9 x sin (5 x^{2} y))

x \to \infty lim (8 - e^{x})

\int \frac{3 ( x - 3 ) ^{2}}{16} d x

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

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