What is (\partial)/(\partial y)(ln((y^9)/(x^7))) ?

The answer to (\partial)/(\partial y)(ln((y^9)/(x^7))) is 9/y

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(\partial)/(\partial y)(ln((y^9)/(x^7)))

\frac{\partial}{\partial y} (ln (\frac{y ^{9}}{x ^{7}}))

\frac{9}{y}

Solution steps

\frac{\partial}{\partial y} (ln (\frac{y ^{9}}{x ^{7}}))

Treat

x

as a constant

Apply the chain rule:

\frac{1}{\frac{y ^{9}}{x ^{7}}} \frac{\partial}{\partial y} (\frac{y ^{9}}{x ^{7}})

= \frac{1}{\frac{y ^{9}}{x ^{7}}} \frac{\partial}{\partial y} (\frac{y ^{9}}{x ^{7}})

\frac{\partial}{\partial y} (\frac{y ^{9}}{x ^{7}}) = \frac{9 y ^{8}}{x ^{7}}

= \frac{1}{\frac{y ^{9}}{x ^{7}}} \cdot \frac{9 y ^{8}}{x ^{7}}

Simplify

\frac{1}{\frac{y ^{9}}{x ^{7}}} \cdot \frac{9 y ^{8}}{x ^{7}} : \frac{9}{y}

= \frac{9}{y}

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What is (\partial)/(\partial y)(ln((y^9)/(x^7))) ?
The answer to (\partial)/(\partial y)(ln((y^9)/(x^7))) is 9/y