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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

f (x) = 2 + 3 x - 5 x^{2}, at x = 1

h \to 1 lim (4 h^{2} + x + 3)

\frac{d}{d x} (e^{x} sin (9 x))

e^{- 2 x} + \frac{1}{3} e^{x}

x \to \infty lim (0 x)

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