Product Rule for Derivatives/Examples/x times Exponential of x times Sine of x

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Examples of Use of Product Rule for Derivatives

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


Let $u = x$.

Let $v = e^x$.

Let $w = \sin x$.

Thus we have:

$y = u v w$

and so:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} + v w \dfrac {\d u} {\d x}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds x e^x \cos x + x \sin x e^x + e^x \sin x \cdot 1\) Power Rule for Derivatives, Derivative of Sine Function, Derivative of Exponential Function
\(\ds \) \(=\) \(\ds e^x \paren {\paren {1 + x} \sin x + x \cos x}\) simplification