What is (\partial)/(\partial y)(4arctan(x/y)) ?

The answer to (\partial)/(\partial y)(4arctan(x/y)) is -(4x)/(x^2+y^2)

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(\partial)/(\partial y)(4arctan(x/y))

\frac{\partial}{\partial y} (4 arctan (\frac{x}{y}))

- \frac{4 x}{x ^{2} + y ^{2}}

Solution steps

\frac{\partial}{\partial y} (4 arctan (\frac{x}{y}))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{\partial}{\partial y} (arctan (\frac{x}{y}))

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}}

= 4 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}})

Simplify

4 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}}) : - \frac{4 x}{x ^{2} + y ^{2}}

= - \frac{4 x}{x ^{2} + y ^{2}}

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What is (\partial)/(\partial y)(4arctan(x/y)) ?
The answer to (\partial)/(\partial y)(4arctan(x/y)) is -(4x)/(x^2+y^2)