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

Examples of Use of Product Rule for Derivatives

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

Proof

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

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $22$.