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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial y} (x y arcsin (yz))

\frac{d}{d x} (e^{- (c o s (x) + s i n (y))})

\frac{d ^{2}}{d x ^{2}} (\frac{x ^{2}}{5 + 4 x})

\int (5 x) d x

\int (1 - cos (x))^{3} sin (x) d x

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