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

Take the constant out:

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

= - \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}

= - \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}}

n = 0 \sum \infty (\frac{5}{4})^{n}

(8.1) (2.6)

cos (a^{4} + x^{4})

\frac{\partial}{\partial x} (\frac{1 x - 1 y}{1 x + 1 y})

n = 0 \sum \infty (- 1)^{n} \frac{( ln ( 2 ) ) ^{n}}{n !}

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