Product Rule for Derivatives/Examples/x squared times Arctangent of x

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Examples of Use of Product Rule for Derivatives

$\map {\dfrac \d {\d x} } {x^2 \arctan x} = 2 x \arctan x + \dfrac {x^2} {1 + x^2}$


Let $u = x^2$.

Let $v = \arctan x$.

Thus we have:

$y = u v$

and so:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds v \dfrac {\d u} {\d x} + u \dfrac {\d v} {\d x}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds \arctan x \cdot 2 x + x^2 \cdot \dfrac 1 {1 + x^2}\) Power Rule for Derivatives, Derivative of Arctangent Function
\(\ds \) \(=\) \(\ds 2 x \arctan x + \dfrac {x^2} {1 + x^2}\) simplification