Product Rule for Derivatives/Examples/x squared times Arctangent of x
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Examples of Use of Product Rule for Derivatives
- $\map {\dfrac \d {\d x} } {x^2 \arctan x} = 2 x \arctan x + \dfrac {x^2} {1 + x^2}$
Proof
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 |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $17$.