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

The answer to (\partial)/(\partial y)((x+y)(x^2+2y^2)) is 6y^2+4xy+x^2

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

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

6 y^{2} + 4 x y + x^{2}

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x + y, g = x^{2} + 2 y^{2}

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

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

\frac{\partial}{\partial y} (x^{2} + 2 y^{2}) = 4 y

= 1 \cdot (x^{2} + 2 y^{2}) + 4 y (x + y)

Simplify

1 \cdot (x^{2} + 2 y^{2}) + 4 y (x + y) : 6 y^{2} + 4 x y + x^{2}

= 6 y^{2} + 4 x y + x^{2}

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What is (\partial)/(\partial y)((x+y)(x^2+2y^2)) ?
The answer to (\partial)/(\partial y)((x+y)(x^2+2y^2)) is 6y^2+4xy+x^2