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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

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

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

(ln (x + 1))^{2} - e^{x^{2}}

\int \frac{( z + 2 )}{z ^{2} + 3 z - 1} d z

\int t sin (π t) d t

\frac{\partial}{\partial x} (5 \cdot (y + 3)^{3} \cdot (x - 4))

f (x) = (x^{2} + 2)^{2}

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