What is (\partial)/(\partial y)(2arctan(x/y)) ?

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

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(\partial)/(\partial y)(2arctan(x/y))

\frac{\partial}{\partial y} (2 arctan (\frac{x}{y}))

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

Solution steps

\frac{\partial}{\partial y} (2 arctan (\frac{x}{y}))

Treat

x

as a constant

Take the constant out:

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

= 2 \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}}

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

Simplify

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

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

2 x^{e}

\frac{d y}{d x} = 4 y^{2} x

\int cot (10 x) d x

\int_{0}^{2} (\frac{x}{2})^{3} d x

\frac{\partial}{\partial x} (a x^{3} + x e^{x} - 1)

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