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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

(x - y) d x = - x d y

- \frac{11}{s ^{6}}

\int_{- 1}^{1} \frac{2 5 - 4 x}{3} d x

\frac{( 7 s ^{2} - 41 s + 84 )}{( s - 1 ) ( s ^{2} - 4 s + 13 )}

\frac{x + 1}{x ^{2} + x + 10}

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