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

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

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(\partial)/(\partial x)((x+y)ln(xy))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

1 \cdot ln (x y) + \frac{1}{x} (x + y) : ln (x y) + \frac{x + y}{x}

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

x \to 0 lim (\frac{4}{x ^{5}})

\frac{d}{d x} (x^{2} sin (x))

\frac{d}{d x} (e^{k t a n (7 x)})

5 - x - x^{2}

\int \frac{5 x}{x ( x - 1 )} d x

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