Derivative of Hyperbolic Tangent

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x = \dfrac 1 {\cosh^2 x}$

where $\tanh$ is the hyperbolic tangent, $\sech$ is the hyperbolic secant and $\cosh$ is the hyperbolic cosine.


Corollary

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


Proof

\(\ds \map {\dfrac \d {\d x} } {\tanh x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\dfrac {\sinh x} {\cosh x} }\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \dfrac {\paren {\dfrac \d {\d x} \sinh x} \cosh x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {\cosh^2 x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}\) Derivative of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \dfrac {\cosh^2 x - \sinh^2 x} {\cosh^2 x}\) Derivative of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \dfrac 1 {\cosh^2 x}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \sech^2 x\) Definition 2 of Hyperbolic Secant

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Derivative_of_Hyperbolic_Tangent&oldid=625643"