Derivative of Hyperbolic Secant/Proof 1
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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 |
$\blacksquare$