What is (\partial)/(\partial x)(x^2(x-y)^{-2}) ?

The answer to (\partial)/(\partial x)(x^2(x-y)^{-2}) is -(2xy)/((x-y)^3)

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(\partial)/(\partial x)(x^2(x-y)^{-2})

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

- \frac{2 x y}{( x - y ) ^{3}}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{2}, g = (x - y)^{- 2}

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

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

\frac{\partial}{\partial x} ((x - y)^{- 2}) = - \frac{2}{( x - y ) ^{3}}

= 2 x (x - y)^{- 2} + (- \frac{2}{( x - y ) ^{3}}) x^{2}

Simplify

2 x (x - y)^{- 2} + (- \frac{2}{( x - y ) ^{3}}) x^{2} : - \frac{2 x y}{( x - y ) ^{3}}

= - \frac{2 x y}{( x - y ) ^{3}}

What is (\partial)/(\partial x)(x^2(x-y)^{-2}) ?
The answer to (\partial)/(\partial x)(x^2(x-y)^{-2}) is -(2xy)/((x-y)^3)