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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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