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

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Examples of Use of Product Rule for Derivatives

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


Proof

Let $u = x$.

Let $v = \cot x$.

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
\(\ds \) \(=\) \(\ds \cot x \cdot 1 + x \cdot \paren {-\cosec^2 x}\) Power Rule for Derivatives, Derivative of Cotangent Function
\(\ds \) \(=\) \(\ds \cot x - x \cosec^2 x\) simplification

$\blacksquare$


Sources