What is (\partial)/(\partial x)((4x+8)ln(6xy)) ?

The answer to (\partial)/(\partial x)((4x+8)ln(6xy)) is 4ln(6xy)+(4x+8)/x

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(\partial)/(\partial x)((4x+8)ln(6xy))

\frac{\partial}{\partial x} ((4 x + 8) ln (6 x y))

4 ln (6 x y) + \frac{4 x + 8}{x}

Solution steps

\frac{\partial}{\partial x} ((4 x + 8) ln (6 x y))

Treat

y

as a constant

Apply the Product Rule:

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

f = 4 x + 8, g = ln (6 x y)

= \frac{\partial}{\partial x} (4 x + 8) ln (6 x y) + \frac{\partial}{\partial x} (ln (6 x y)) (4 x + 8)

\frac{\partial}{\partial x} (4 x + 8) = 4

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

= 4 ln (6 x y) + \frac{1}{x} (4 x + 8)

\frac{1}{x} (4 x + 8) = \frac{4 x + 8}{x}

= 4 ln (6 x y) + \frac{4 x + 8}{x}

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

\frac{d}{d x} (- cos (2 x) - 2 sin (2 x) (1 - x))

x \frac{d y}{d x} + 2 y = x^{2}

x \to 0 lim (\frac{x}{sin ^{2} ( x )})

\frac{d}{d x} (\frac{x}{8 x + 3})

What is (\partial)/(\partial x)((4x+8)ln(6xy)) ?
The answer to (\partial)/(\partial x)((4x+8)ln(6xy)) is 4ln(6xy)+(4x+8)/x