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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (\frac{x}{y}) = \frac{1}{y}

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

Simplify

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

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

\int sin (x - 2) d x

\frac{18}{x}

\int \frac{x ^{3}}{x ^{2} + 2} d x

\int e^{- (x^{2} + 1)} d x

y^{^{''}} + 4 y = cos (4 x), y (0) = 0, y^{^{'}} (0) = 0

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