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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

3 e^{t}

\int csc^{2} (θ) d θ

f (θ) = cot (θ)

\int \frac{1}{u ^{2} + 4} d u

x \to 0 + lim ((5 x)^{x})

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