What is the d/(dy)((x^3y-xy^3)/(x^2+y^2)) ?

The d/(dy)((x^3y-xy^3)/(x^2+y^2)) is (x^5-4x^3y^2-xy^4)/((x^2+y^2)^2)

What is the first d/(dy)((x^3y-xy^3)/(x^2+y^2)) ?

The first d/(dy)((x^3y-xy^3)/(x^2+y^2)) is (x^5-4x^3y^2-xy^4)/((x^2+y^2)^2)

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

\frac{x ^{5} - 4 x ^{3} y ^{2} - x y ^{4}}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Expand

(x^{3} - 3 x y^{2}) (x^{2} + y^{2}) - 2 y (x^{3} y - x y^{3}) : x^{5} - 4 x^{3} y^{2} - x y^{4}

= \frac{x ^{5} - 4 x ^{3} y ^{2} - x y ^{4}}{( x ^{2} + y ^{2} ) ^{2}}

\int e^{x^{- \frac{1}{2}}} d x

\int \frac{1}{2 x ^{2} - 4 x + 14} d x

\int x 5^{x} d x

\frac{d}{d x} (e e^{x} e^{x^{2}})

\int_{0}^{1} \frac{8 + x}{1 + x ^{2}} d x

What is the d/(dy)((x^3y-xy^3)/(x^2+y^2)) ?
The d/(dy)((x^3y-xy^3)/(x^2+y^2)) is (x^5-4x^3y^2-xy^4)/((x^2+y^2)^2)
What is the first d/(dy)((x^3y-xy^3)/(x^2+y^2)) ?
The first d/(dy)((x^3y-xy^3)/(x^2+y^2)) is (x^5-4x^3y^2-xy^4)/((x^2+y^2)^2)