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

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

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

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

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

Solution steps

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

Simplify

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

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

Apply the Quotient Rule:

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

= - y \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

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

Simplify

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

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

x \to 3 + lim (\frac{3 - x}{x ^{2} - 9})

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

\frac{d ^{2}}{d x ^{2}} f (x) = x^{3} + 3 x^{2} - 2 x + 1

\frac{\partial}{\partial x} (2 x y e^{x z})

x

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