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

The answer to (\partial)/(\partial y)((2xy^2)/(2+x^2y^2)) is (8xy)/((2+x^2y^2)^2)

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

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

\frac{8 x y}{( 2 + x ^{2} y ^{2} ) ^{2}}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

2 x \frac{2 y ( 2 + x ^{2} y ^{2} ) - 2 x ^{2} y y ^{2}}{( 2 + x ^{2} y ^{2} ) ^{2}} : \frac{8 x y}{( 2 + x ^{2} y ^{2} ) ^{2}}

= \frac{8 x y}{( 2 + x ^{2} y ^{2} ) ^{2}}

\int \frac{82 x + 8}{( 9 x + 1 ) ( x - 1 )} d x

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

\int \frac{2 x + 1}{3 x ^{2} + 3 x - 2} d x

What is (\partial)/(\partial y)((2xy^2)/(2+x^2y^2)) ?
The answer to (\partial)/(\partial y)((2xy^2)/(2+x^2y^2)) is (8xy)/((2+x^2y^2)^2)