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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\int (x)^{2} d x

\int \frac{1}{x ln ( x ) ( ln ( ln ( x ) ) ) ^{2}} d x

(1 + x^{2} + y^{2} + x^{2} y^{2}) d y = (1 + y^{2}) d x

\int \frac{100}{100 - y} d y

y^{^{'''}} + 4 \cdot y^{^{'}} = 3 x - 1

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