Derivative of Hyperbolic Cotangent

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x = \dfrac {-1} {\sinh^2 x}$

where $\coth x$ denotes the hyperbolic cotangent, $\csch x$ denotes the hyperbolic cosecant and $\sinh x$ denotes the hyperbolic sine.


Corollary

$\map {\dfrac \d {\d x} } {\coth x} = 1 - \coth^2 x$


Proof

\(\ds \map {\dfrac \d {\d x} } {\coth x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }\) Definition 2 of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {\sinh x \sinh x - \cosh x \dfrac \d {\d x} \cosh x} {\sinh^2 x}\) Derivative of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \frac {\sinh x \sinh x - \cosh x \cosh x} {\sinh^2 x}\) Derivative of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {-1} {\sinh^2 x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds -\csch^2 x\) Definition 2 of Hyperbolic Cosecant

$\blacksquare$


Also see


Sources