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

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

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(\partial)/(\partial y)(arctan(x/y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial y} (5 arctan (\frac{x}{y}))

\frac{d y}{d x} = 8 y x

y^{^{'}} = 5 x 3 y + 11, y (0) = 3

f (x) = ln (ln (3 x))

\int \frac{1}{3} x^{- \frac{2}{3}} d x

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