What is (\partial)/(\partial x)(tan(y/x)) ?

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

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(\partial)/(\partial x)(tan(y/x))

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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