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

The answer to (\partial)/(\partial x)(x(x+y)^2) is 3x^2+4xy+y^2

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

1 \cdot (x + y)^{2} + 2 (x + y) x : 3 x^{2} + 4 x y + y^{2}

= 3 x^{2} + 4 x y + y^{2}

\frac{d}{d x} (\frac{4 x ^{3} + 9}{x})

\frac{d}{d t} (sin (ω t + θ))

\int_{0}^{1} (1) d x

x e^{x} y^{^{'}} + 2 e^{x} y = 0

What is (\partial)/(\partial x)(x(x+y)^2) ?
The answer to (\partial)/(\partial x)(x(x+y)^2) is 3x^2+4xy+y^2