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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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