What is the d/(dy)((x^2+y^2)arctan(y/x)) ?

The d/(dy)((x^2+y^2)arctan(y/x)) is 2yarctan(y/x)+x

What is the first d/(dy)((x^2+y^2)arctan(y/x)) ?

The first d/(dy)((x^2+y^2)arctan(y/x)) is 2yarctan(y/x)+x

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = x^{2} + y^{2}, g = arctan (\frac{y}{x})

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

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

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

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

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

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

What is the d/(dy)((x^2+y^2)arctan(y/x)) ?
The d/(dy)((x^2+y^2)arctan(y/x)) is 2yarctan(y/x)+x
What is the first d/(dy)((x^2+y^2)arctan(y/x)) ?
The first d/(dy)((x^2+y^2)arctan(y/x)) is 2yarctan(y/x)+x