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

The answer to (\partial)/(\partial x)((x+y)ln(x+y)) is ln(x+y)+1

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

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

ln (x + y) + 1

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 + y}

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

Simplify

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

= ln (x + y) + 1

What is (\partial)/(\partial x)((x+y)ln(x+y)) ?
The answer to (\partial)/(\partial x)((x+y)ln(x+y)) is ln(x+y)+1