Derivative of Cotangent Function/Corollary 3

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Corollary to Derivative of Cotangent Function

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


Proof

\(\ds \map {\dfrac \d {\d x} } {\cot x}\) \(=\) \(\ds -\csc^2 x\) Derivative of $\cot x$
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\cot a x}\) \(=\) \(\ds -a \csc^2 a x\) Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds -a \paren {\cot^2 a x + 1}\) Difference of Squares of Cosecant and Cotangent

$\blacksquare$


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