What is the derivative of xarctan(x) ?

The derivative of xarctan(x) is arctan(x)+x/(x^2+1)

What is the first derivative of xarctan(x) ?

The first derivative of xarctan(x) is arctan(x)+x/(x^2+1)

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\frac{d}{d x} (x arctan (x))

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

Solution steps

\frac{d}{d x} (x arctan (x))

Apply the Product Rule:

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

f = x, g = arctan (x)

= \frac{d x}{d x} arctan (x) + \frac{d}{d x} (arctan (x)) x

\frac{d x}{d x} = 1

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

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

Simplify

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

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

What is the derivative of xarctan(x) ?
The derivative of xarctan(x) is arctan(x)+x/(x^2+1)
What is the first derivative of xarctan(x) ?
The first derivative of xarctan(x) is arctan(x)+x/(x^2+1)