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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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