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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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