What is (\partial)/(\partial x)(0.5xarctan(y/x)) ?

The answer to (\partial)/(\partial x)(0.5xarctan(y/x)) is 0.5(arctan(y/x)-(xy)/(y^2+x^2))

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(\partial)/(\partial x)(0.5xarctan(y/x))

\frac{\partial}{\partial x} (0.5 x arctan (\frac{y}{x}))

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

Solution steps

\frac{\partial}{\partial x} (0.5 x arctan (\frac{y}{x}))

Treat

y

as a constant

Take the constant out:

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

= 0.5 \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})

= 0.5 (\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}}

= 0.5 (1 \cdot arctan (\frac{y}{x}) + (- \frac{y}{y ^{2} + x ^{2}}) x)

Simplify

0.5 (1 \cdot arctan (\frac{y}{x}) + (- \frac{y}{y ^{2} + x ^{2}}) x) : 0.5 (arctan (\frac{y}{x}) - \frac{x y}{y ^{2} + x ^{2}})

= 0.5 (arctan (\frac{y}{x}) - \frac{x y}{y ^{2} + x ^{2}})

\int x^{2} sin (\frac{x}{2}) d x

\int \frac{1 + x}{x} d x

2 \frac{d y}{d x} - 4 x y = 8 x, y (0) = 2

\int 6 arctan (\frac{8}{x}) d x

y = x^{0.3}

What is (\partial)/(\partial x)(0.5xarctan(y/x)) ?
The answer to (\partial)/(\partial x)(0.5xarctan(y/x)) is 0.5(arctan(y/x)-(xy)/(y^2+x^2))