Derivative of Inverse Hyperbolic Sine of x over a

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Theorem

$\dfrac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$


Corollary 1

$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$


Corollary 2

$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$


Proof

\(\displaystyle \frac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x}\) \(=\) \(\displaystyle \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}\) Derivative of $\sinh^{-1}$ and Derivative of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \frac a {\sqrt {x^2 + a^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\sqrt {x^2 + a^2} }\)

$\blacksquare$


Also see


Sources