What is (\partial)/(\partial y)(y/((x+y)^2)) ?

The answer to (\partial)/(\partial y)(y/((x+y)^2)) is (-y+x)/((x+y)^3)

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(\partial)/(\partial y)(y/((x+y)^2))

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

\frac{- y + x}{( x + y ) ^{3}}

Solution steps

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

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

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

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

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

Simplify

\frac{1 \cdot ( x + y ) ^{2} - 2 ( x + y ) y}{( ( x + y ) ^{2} ) ^{2}} : \frac{- y + x}{( x + y ) ^{3}}

= \frac{- y + x}{( x + y ) ^{3}}

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

f (t) = t^{\frac{5}{3}} e^{t}

\int_{0}^{\infty} e^{- x} cos (x) d x

\frac{\partial}{\partial x} (e^{- 2 t} cos (π x))

\frac{d y}{d t} = \frac{t}{y e ^{y + t^{2}}}

What is (\partial)/(\partial y)(y/((x+y)^2)) ?
The answer to (\partial)/(\partial y)(y/((x+y)^2)) is (-y+x)/((x+y)^3)