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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

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

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

x \to 2 - lim (\frac{1}{( x - 2 ) ^{2}})

\frac{d y}{d x} = 4 x - 5 y

2 x^{2} + 5 x + 1

cos^{2} (t) y^{^{'}} = 1

\int \frac{1}{e ^{y}} d y

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