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

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

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

\frac{\partial}{\partial x} (\frac{x \cdot 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}})

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}}

\int x (sin (x) - cos (x^{2})) d x

y^{^{''}} + 4 y = 3 sin (x)

\frac{d V}{12 - 2 V} = d t, V (0) = 1

\frac{\partial}{\partial x} (\frac{x + 2 y}{3 x + 4 y})

\frac{d}{d θ} (cos (3 θ))

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