Product Rule for Derivatives/Examples/Cotangent of x times Exponential of -x

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Examples of Use of Product Rule for Derivatives

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


Let $u = \cot x$.

Let $v = e^{-x}$.

Thus we have:

$y = u v$

and so:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds \cot x \paren {-e^{-x} } + e^{-x} \paren {-\cosec^2 x}\) Derivative of Exponential Function: Corollary 3, Derivative of Cotangent Function
\(\ds \) \(=\) \(\ds -e^{-x} \paren {\cot x + \cosec^2 x}\) simplification