What is (\partial}{\partial y}(\frac{ln(xy))/y) ?

The answer to (\partial}{\partial y}(\frac{ln(xy))/y) is (1-ln(xy))/(y^2)

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(\partial}{\partial y}(\frac{ln(xy))/y)

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

\frac{1 - ln ( x y )}{y ^{2}}

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial y} ( ln ( x y ) ) y - \frac{\partial}{\partial y} ( y ) ln ( x y )}{y ^{2}}

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

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

= \frac{\frac{1}{y} y - 1 \cdot ln ( x y )}{y ^{2}}

Simplify

\frac{\frac{1}{y} y - 1 \cdot ln ( x y )}{y ^{2}} : \frac{1 - ln ( x y )}{y ^{2}}

= \frac{1 - ln ( x y )}{y ^{2}}

What is (\partial}{\partial y}(\frac{ln(xy))/y) ?
The answer to (\partial}{\partial y}(\frac{ln(xy))/y) is (1-ln(xy))/(y^2)