What is (\partial)/(\partial y)(4arctan(x/y)) ?

The answer to (\partial)/(\partial y)(4arctan(x/y)) is -(4x)/(x^2+y^2)

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(\partial)/(\partial y)(4arctan(x/y))

\frac{\partial}{\partial y} (4 arctan (\frac{x}{y}))

- \frac{4 x}{x ^{2} + y ^{2}}

Solution steps

\frac{\partial}{\partial y} (4 arctan (\frac{x}{y}))

Treat

x

as a constant

Take the constant out:

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

= 4 \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}}

= 4 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}})

Simplify

4 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}}) : - \frac{4 x}{x ^{2} + y ^{2}}

= - \frac{4 x}{x ^{2} + y ^{2}}

\frac{d}{d x} (\frac{x ^{2} - 2 x - 48}{x + 6})

f (x) = 4 x^{2} - 2 x + 3

tan (3 x) + x^{4}

y^{^{''}} + 11 y^{^{'}} + 28 y = 84 x^{2} + 66 x + 6 + 40 e^{x}

x \to 12 lim (x - 3)

What is (\partial)/(\partial y)(4arctan(x/y)) ?
The answer to (\partial)/(\partial y)(4arctan(x/y)) is -(4x)/(x^2+y^2)