Stan Math Library
5.0.0
Automatic Differentiation
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The inverse hyperbolic cosine function for variables (C99).
For non-variable function, see acosh().
The derivative is defined by
\(\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}\).
\[ \mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]
\[ \frac{\partial\, \mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]
\[ \cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \]
\[ \frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}} \]
x | The variable. |