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

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

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(\partial)/(\partial y)((e^y)/(e^x+e^y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{d}{d x} (3 x^{2} - 2 x + 5)

\int (x y e^{x z}) d z

\int x I n x d x

\int - \frac{1}{( x - 1 ) ^{2}} d x

\frac{3 w ^{6} - w}{w}

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