What is (\partial)/(\partial x)((2x+8)ln(xy)) ?

The answer to (\partial)/(\partial x)((2x+8)ln(xy)) is 2ln(xy)+(2x+8)/x

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(\partial)/(\partial x)((2x+8)ln(xy))

\frac{\partial}{\partial x} ((2 x + 8) ln (x y))

2 ln (x y) + \frac{2 x + 8}{x}

Solution steps

\frac{\partial}{\partial x} ((2 x + 8) ln (x y))

Treat

y

as a constant

Apply the Product Rule:

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

f = 2 x + 8, g = ln (x y)

= \frac{\partial}{\partial x} (2 x + 8) ln (x y) + \frac{\partial}{\partial x} (ln (x y)) (2 x + 8)

\frac{\partial}{\partial x} (2 x + 8) = 2

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

= 2 ln (x y) + \frac{1}{x} (2 x + 8)

\frac{1}{x} (2 x + 8) = \frac{2 x + 8}{x}

= 2 ln (x y) + \frac{2 x + 8}{x}

What is (\partial)/(\partial x)((2x+8)ln(xy)) ?
The answer to (\partial)/(\partial x)((2x+8)ln(xy)) is 2ln(xy)+(2x+8)/x