Step By Step Calculus » 10.4 - Derivatives for Hyperbolic and Inverse Functions

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Synopsis
The following table shows the derivatives of 6 hyperbolic functions and 6 inverse hyperbolic functions.
\newcommand{\T}{\rule{0pt}{2.2ex}} \newcommand{\B}{\rule[-1.4ex]{0pt}{0pt}} \begin{tabular}{|l|c|c||l|c|c|} \hline Function & Derivative & $\begin{array}{c}\textrm{Derivative}\\\textrm{Domain}\end{array}$ & Function & Derivative & $\begin{array}{c}\textrm{Derivative}\\\textrm{Domain}\end{array}$ \\ \hline \T\B$\sinh x$ & $\cosh x$ & $\mathbb{R}$ & $\cosh x$ & $\sinh x$ & $\mathbb{R}$\\ \hline \T\B$\tanh x$ & $\func{sech}^{2}x$ & $\mathbb{R}$& $\coth x$ & $-\func{csch}^{2}x$ & $\mathbb{R}-\{0\}$\\ \hline \T\B$\func{sech} x$ & $-\func{sech} x\tanh x$ & $\mathbb{R}$& $\func{csch} x$ & $-\func{csch}x\coth x$ & $\mathbb{R}-\{0\}$\\ \hline \T\B$\func{sinh}^{-1}x$ & $\frac{1}{\sqrt{1+x^{2}}}$ & $\mathbb{R}$& \T\B$\func{cosh}^{-1}x$ & $\frac{1}{\sqrt{x^{2}-1}}$ & $(1,\infty)$\\ \hline \T\B$\func{tanh}^{-1}x$ & $\frac{1}{1-x^{2}}$ & $(-1,1)$& \T\B$\func{coth}^{-1}x$ & $\frac{1}{1-x^{2}}$ & $[-1,1]^C$\\ \hline \T\B$\func{sech}^{-1}x$ & $-\frac{1}{x\sqrt{1-x^{2}}}$ & $(0,1)$& \T\B$\func{csch}^{-1}x$ & $-\frac{1}{\left| x\right| \sqrt{x^{2}+1}}$ & $\mathbb{R}-\{0\}$\\ \hline \end{tabular}