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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (sin (e^{x} (x^{2} - x)))

2 x^{2} - 2 x - 4, at x = 1

\frac{cos ( x )}{x ^{5}}

\frac{d}{d x} (\frac{x}{6 - x})

y = 3 x^{2} + 12 x + 6

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