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

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

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(\partial)/(\partial x)(xyln(xy))

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

y (ln (x y) + 1)

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

f = x, g = ln (x y)

= 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}

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

Simplify

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

= y (ln (x y) + 1)

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