What is (\partial)/(\partial x)(xy(6-x-y)) ?

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

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(\partial)/(\partial x)(xy(6-x-y))

\frac{\partial}{\partial x} (x y (6 - x - y))

y (- 2 x - y + 6)

Solution steps

\frac{\partial}{\partial x} (x y (6 - x - y))

Treat

y

as a constant

Take the constant out:

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

= y \frac{\partial}{\partial x} (x (6 - x - y))

Apply the Product Rule:

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

f = x, g = 6 - x - y

= y (\frac{\partial}{\partial x} (x) (6 - x - y) + \frac{\partial}{\partial x} (6 - x - y) x)

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

\frac{\partial}{\partial x} (6 - x - y) = - 1

= y (1 \cdot (6 - x - y) + (- 1) x)

Simplify

y (1 \cdot (6 - x - y) + (- 1) x) : y (- 2 x - y + 6)

= y (- 2 x - y + 6)

\frac{d}{d x} (x^{\frac{4}{3}} - 4 x^{\frac{1}{3}} + 1)

\frac{\partial}{\partial x} (cos (4 y - x))

\frac{\partial}{\partial x} (4 y^{x})

9 x - 9 x

\frac{d}{d x} ((x^{2} + y^{2}) sin (\frac{1}{x ^{2} + y ^{2}}))

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