What is (\partial)/(\partial y)((xy)/((x-y))) ?

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

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(\partial)/(\partial y)((xy)/((x-y)))

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

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

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

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

Simplify

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

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

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