What is (\partial)/(\partial x)((ye^{-y/x})/(x^2)) ?

The answer to (\partial)/(\partial x)((ye^{-y/x})/(x^2)) is (y(e^{-y/x}y-2e^{-y/x}x))/(x^4)

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(\partial)/(\partial x)((ye^{-y/x})/(x^2))

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

\frac{y ( e ^{- \frac{y}{x}} y - 2 e ^{- \frac{y}{x}} x )}{x ^{4}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

y \frac{\frac{y e ^{- \frac{y}{x}}}{x ^{2}} x ^{2} - 2 x e ^{- \frac{y}{x}}}{( x ^{2} ) ^{2}} : \frac{y ( e ^{- \frac{y}{x}} y - 2 e ^{- \frac{y}{x}} x )}{x ^{4}}

= \frac{y ( e ^{- \frac{y}{x}} y - 2 e ^{- \frac{y}{x}} x )}{x ^{4}}

\int - 6 y d y

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

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

f (x) = \frac{x}{x - 3}

f (x) = \frac{x ^{2} + x - 2}{x ^{3} + 6}

What is (\partial)/(\partial x)((ye^{-y/x})/(x^2)) ?
The answer to (\partial)/(\partial x)((ye^{-y/x})/(x^2)) is (y(e^{-y/x}y-2e^{-y/x}x))/(x^4)