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

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

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

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

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

Solution steps

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

Treat

y

as a constant

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}

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

Simplify

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

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

\frac{d}{d x} (4 x^{3} - 12 x)

\int_{3}^{5} \frac{2 x ^{2} + x + 4}{x - 1} d x

\int_{1}^{e} \frac{( 1 + ln ( x ) ) ^{2}}{x} d x

\int \frac{tan ( ln ( x ) )}{x} d x

(3 x^{3} + x^{- 3}) (x + 3) (x^{2} - 5)

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