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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

= - y^{2} (- \frac{2}{( x - y ) ^{3}} \cdot 1)

Simplify

- y^{2} (- \frac{2}{( x - y ) ^{3}} \cdot 1) : \frac{2 y ^{2}}{( x - y ) ^{3}}

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

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