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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

= 2 (\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}}

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

Simplify

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

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

ln (x^{2} + 2)

x \to - 5 lim (x + 6)

\frac{d}{d x} (\frac{x cos ( x ) + π}{π - x})

\int \frac{1}{e ^{2 x} - 1} d x

\int \frac{x ^{2}}{1 + x} d x

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