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 |
$\blacksquare$