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}}

\frac{d}{d x} (x^{2} - \frac{4}{x ^{3}})

\frac{d}{d x} (\frac{8 x ^{2} + 4 x + 2}{x})

\int_{1}^{\infty} \frac{x}{e ^{x}} d x

\frac{\partial}{\partial x} (60 x^{\frac{4}{5}} y^{\frac{1}{5}})

\int \frac{x}{4 - 9 x ^{2}} d x

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