What is (\partial)/(\partial x)(1/2 xarctan(y/x)) ?

The answer to (\partial)/(\partial x)(1/2 xarctan(y/x)) is 1/2 (arctan(y/x)-(xy)/(y^2+x^2))

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(\partial)/(\partial x)(1/2 xarctan(y/x))

\frac{\partial}{\partial x} (\frac{1}{2} x arctan (\frac{y}{x}))

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

Solution steps

\frac{\partial}{\partial x} (\frac{1}{2} x arctan (\frac{y}{x}))

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = arctan (\frac{y}{x})

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

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

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

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

Simplify

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

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

\int \frac{csc ^{2} ( x )}{4} d x

\int 4 cos (2 x) d x

\int (2 x - 3)^{- 1} (x + 1)^{- 1} d x

3 x^{3} + 9 x y^{2} = 5 x^{3}

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

What is (\partial)/(\partial x)(1/2 xarctan(y/x)) ?
The answer to (\partial)/(\partial x)(1/2 xarctan(y/x)) is 1/2 (arctan(y/x)-(xy)/(y^2+x^2))