Product Rule for Derivatives/Examples

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Examples of Use of Product Rule for Derivatives

Example: $2 a x e^{a x^2}$

$\map {\dfrac \d {\d x} } {2 a x e^{a x^2} } = 2 a e^{a x^2} \paren {1 + 2 a x^2}$

Example: $x \cot x$

$\map {\dfrac \d {\d x} } {x \cot x} = \cot x - x \cosec^2 x$

Example: $x^2 \arctan x$

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

Example: $x e^x \sin x$

$\map {\dfrac \d {\d x} } {x e^x \sin x} = e^x \paren {\paren {1 + x} \sin x + x \cos x}$

Example: $\cot x e^{-x}$

$\map {\dfrac \d {\d x} } {\cot x e^{-x} } = -e^{-x} \paren {\cot x + \cosec^2 x}$