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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{d}{d x} (- \frac{2}{( 6 x - 7 x ^{4} ) ^{3}})

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\int_{6}^{12} 2 x - 18 d x

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