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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int e^{- 0.5 x^{2}} d x

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

\int 5^{t a n (x)} sec^{2} (x) d x

\frac{d y}{d x} = \frac{4 x + x y ^{2}}{y - x ^{2} y}

e^{8 e^{x}} \cdot 8 e^{x}

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