What is (\partial)/(\partial y)(ln((y/x)^2)) ?

The answer to (\partial)/(\partial y)(ln((y/x)^2)) is 2/y

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(\partial)/(\partial y)(ln((y/x)^2))

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

\frac{2}{y}

Solution steps

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

Simplify

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

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

Treat

x

as a constant

Apply the chain rule:

\frac{1}{\frac{y ^{2}}{x ^{2}}} \frac{\partial}{\partial y} (\frac{y ^{2}}{x ^{2}})

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

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

= \frac{1}{\frac{y ^{2}}{x ^{2}}} \cdot \frac{2 y}{x ^{2}}

Simplify

\frac{1}{\frac{y ^{2}}{x ^{2}}} \cdot \frac{2 y}{x ^{2}} : \frac{2}{y}

= \frac{2}{y}

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What is (\partial)/(\partial y)(ln((y/x)^2)) ?
The answer to (\partial)/(\partial y)(ln((y/x)^2)) is 2/y