What is (\partial)/(\partial x)(e^{x/y}(1-x/y)) ?

The answer to (\partial)/(\partial x)(e^{x/y}(1-x/y)) is -(e^{x/y}x)/(y^2)

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(\partial)/(\partial x)(e^{x/y}(1-x/y))

\frac{\partial}{\partial x} (e^{\frac{x}{y}} (1 - \frac{x}{y}))

- \frac{e ^{\frac{x}{y}} x}{y ^{2}}

Solution steps

\frac{\partial}{\partial x} (e^{\frac{x}{y}} (1 - \frac{x}{y}))

Treat

y

as a constant

Apply the Product Rule:

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

f = e^{\frac{x}{y}}, g = 1 - \frac{x}{y}

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial x)(e^{x/y}(1-x/y)) ?
The answer to (\partial)/(\partial x)(e^{x/y}(1-x/y)) is -(e^{x/y}x)/(y^2)