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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\int \frac{5 ln ^{2} ( x ) + 9}{x} d x

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

\int \frac{1}{5 sin ( x ) + 3} d x

\frac{\partial}{\partial x} (x^{2} + y^{2} + 3 x y + 15 x)

\frac{d}{d t} (\frac{e ^{t} θ}{( 1 - e ^{t} ( 1 - θ ) ) ^{2}})

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