What is (\partial)/(\partial y)((xy+3x^2)/(1+y)) ?

The answer to (\partial)/(\partial y)((xy+3x^2)/(1+y)) is (x-3x^2)/((1+y)^2)

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(\partial)/(\partial y)((xy+3x^2)/(1+y))

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

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

Solution steps

\frac{\partial}{\partial y} (\frac{x y + 3 x ^{2}}{1 + y})

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

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

\int \frac{9 x ^{2} + 4}{x ^{2}} d x

What is (\partial)/(\partial y)((xy+3x^2)/(1+y)) ?
The answer to (\partial)/(\partial y)((xy+3x^2)/(1+y)) is (x-3x^2)/((1+y)^2)