What is (\partial)/(\partial y)((3xy)/(x^2+y^2)) ?

The answer to (\partial)/(\partial y)((3xy)/(x^2+y^2)) is (3x(x^2-y^2))/((x^2+y^2)^2)

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(\partial)/(\partial y)((3xy)/(x^2+y^2))

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

= 3 x \frac{1 \cdot ( x ^{2} + y ^{2} ) - 2 yy}{( x ^{2} + y ^{2} ) ^{2}}

Simplify

3 x \frac{1 \cdot ( x ^{2} + y ^{2} ) - 2 yy}{( x ^{2} + y ^{2} ) ^{2}} : \frac{3 x ( x ^{2} - y ^{2} )}{( x ^{2} + y ^{2} ) ^{2}}

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

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

cos (x) d y + (y sin (x) - cos^{4} (x)) d x = 0

f (x) = x + 2 sin (x)

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

2 x y d x + x^{2} d y = 0

What is (\partial)/(\partial y)((3xy)/(x^2+y^2)) ?
The answer to (\partial)/(\partial y)((3xy)/(x^2+y^2)) is (3x(x^2-y^2))/((x^2+y^2)^2)