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

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

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

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

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

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

Solution steps

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

\frac{d}{d y} (x^{2} y^{2} + 2 x y^{3}) = 2 x^{2} y + 6 x y^{2}

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

= \frac{( 2 x ^{2} y + 6 x y ^{2} ) ( x + y ) ^{2} - 2 ( x + y ) ( x ^{2} y ^{2} + 2 x y ^{3} )}{( ( x + y ) ^{2} ) ^{2}}

Simplify

\frac{( 2 x ^{2} y + 6 x y ^{2} ) ( x + y ) ^{2} - 2 ( x + y ) ( x ^{2} y ^{2} + 2 x y ^{3} )}{( ( x + y ) ^{2} ) ^{2}} : \frac{2 x y ( x ^{2} + 3 x y + y ^{2} )}{( x + y ) ^{3}}

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

\int x sin (x) d x

\frac{d}{d x} (\frac{2 - sec ( x )}{tan ( x )})

y = 2 - 3 x, x = - 2, x = 2, y = 0

\int_{- 2}^{2} x^{3} d x

y = 4 x

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