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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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