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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

x \to \infty lim (x + x^{3})

\int e^{- s^{2}} d s

h \to 0 lim (\frac{2. 7 ^{h} - 1}{h})

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

\int \frac{1}{- x ^{2} + 12 x - 32} d x

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