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}}

x \to 5 - lim (\frac{2}{x})

x \to 2 lim (\frac{x ^{2} + 6 x - 2 x}{x - 2})

\frac{1}{x ^{5}} \frac{d y}{d x} = 2 + 18 y

cos^{8} (x)

- ln (cos (t))

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)