What is (\partial)/(\partial x)(xln(xy)) ?

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

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(\partial)/(\partial x)(xln(xy))

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

ln (x y) + 1

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (x y)

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

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

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

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

Simplify

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

= ln (x y) + 1

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