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}})

Simplify

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

= \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}}

y^{^{''}} = a y

y^{^{'''}} - 4 y^{^{''}} - 5 y^{^{'}} = 0

x y = - 100

\int 10 x \cdot x d x

f (x) = 3 x - 8

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)