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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{x}{4 + x ^{2}}

y^{^{'}} + y = x y^{3}

y = (\frac{x ^{2} + 3}{x ^{2} - 3})^{4}

x \to 1 lim (- x)

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

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