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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (5 x^{3} y + 3 sin (x y))

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

\int x \cdot cos (x^{2} + 1) d x

\int \frac{3 x - 1}{4 x ^{2} - 4 x + 17} d x

\frac{d y}{d x} = \frac{( 3 - y )}{2 x}

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