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

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

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

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

- \frac{2}{x}

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (\frac{y ^{2}}{x ^{2}}) = - \frac{2 y ^{2}}{x ^{3}}

= \frac{1}{\frac{y ^{2}}{x ^{2}}} (- \frac{2 y ^{2}}{x ^{3}})

Simplify

\frac{1}{\frac{y ^{2}}{x ^{2}}} (- \frac{2 y ^{2}}{x ^{3}}) : - \frac{2}{x}

= - \frac{2}{x}

\int \frac{x ^{2} - 8 x - 17}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

\frac{\partial}{\partial x} (41 - x^{2} - y^{2})

\frac{\partial}{\partial x} (sin (2 y))

x \to 1 lim (∣ x ∣)

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