What is (\partial)/(\partial y)(yzln(xy)) ?

The answer to (\partial)/(\partial y)(yzln(xy)) is z(ln(yx)+1)

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(\partial)/(\partial y)(yzln(xy))

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

z (ln (y x) + 1)

Solution steps

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

Treat

z, x

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

f = y, g = ln (x y)

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

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

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

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

Simplify

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

= z (ln (y x) + 1)

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What is (\partial)/(\partial y)(yzln(xy)) ?
The answer to (\partial)/(\partial y)(yzln(xy)) is z(ln(yx)+1)