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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

= \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}}

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

Simplify

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

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

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