Derivative of Hyperbolic Secant/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$


Proof

\(\ds \map {\frac \d {\d x} } {\sech x}\) \(=\) \(\ds \map {\frac \d {\d x} } {\frac 1 {\cosh x} }\) Definition of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \map {\frac \d {\d x} } {\paren {\cosh x}^{-1} }\) Exponent Laws
\(\ds \) \(=\) \(\ds -\paren {\cosh x}^{-2} \sinh x\) Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {-1} {\cosh x} \frac {\sinh x} {\cosh x}\) Exponent Combination Laws
\(\ds \) \(=\) \(\ds -\sech z \tanh z\) Definition of Hyperbolic Secant and Definition of Hyperbolic Tangent

$\blacksquare$