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

The answer to (\partial)/(\partial x)((x+y)(2x+y)^2) is 12x^2+16xy+5y^2

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

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

12 x^{2} + 16 x y + 5 y^{2}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

1 \cdot (2 x + y)^{2} + 4 (2 x + y) (x + y) : 12 x^{2} + 16 x y + 5 y^{2}

= 12 x^{2} + 16 x y + 5 y^{2}

What is (\partial)/(\partial x)((x+y)(2x+y)^2) ?
The answer to (\partial)/(\partial x)((x+y)(2x+y)^2) is 12x^2+16xy+5y^2