What is (\partial)/(\partial x)(xy(x-y)^{-1}) ?

The answer to (\partial)/(\partial x)(xy(x-y)^{-1}) is -(y^2)/((x-y)^2)

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(\partial)/(\partial x)(xy(x-y)^{-1})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = (x - y)^{- 1}

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial x)(xy(x-y)^{-1}) ?
The answer to (\partial)/(\partial x)(xy(x-y)^{-1}) is -(y^2)/((x-y)^2)