What is (\partial)/(\partial x)(ln(x/y+y/x)) ?

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

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(\partial)/(\partial x)(ln(x/y+y/x))

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

\frac{x ^{2} - y ^{2}}{x ( x ^{2} + y ^{2} )}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

\frac{1}{\frac{x}{y} + \frac{y}{x}} \frac{\partial}{\partial x} (\frac{x}{y} + \frac{y}{x})

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

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

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

Simplify

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

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

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