What is (\partial)/(\partial y)(xy(1-x-y)) ?

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

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(\partial)/(\partial y)(xy(1-x-y))

\frac{\partial}{\partial y} (x y (1 - x - y))

x (1 - x - 2 y)

Solution steps

\frac{\partial}{\partial y} (x y (1 - x - y))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

f = y, g = 1 - x - y

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

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

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

= x (1 \cdot (1 - x - y) + (- 1) y)

Simplify

x (1 \cdot (1 - x - y) + (- 1) y) : x (1 - x - 2 y)

= x (1 - x - 2 y)

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

x \to 9 lim (∣ x + 3 ∣)

3 t \cdot \frac{d y}{d t} - y = t

\int x + e^{4 x} d x

\frac{\partial}{\partial x} (3 a x^{2} + e^{x} x + e^{x})

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