Derivative of Inverse Hyperbolic Sine of x over a

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Theorem

$\dfrac {\mathrm d \left({\sinh^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$


Proof

\(\displaystyle \frac {\mathrm d \left({\sinh^{-1} \frac x a}\right)} {\mathrm d x}\) \(=\) \(\displaystyle \frac 1 a \frac 1 {\sqrt {\left({\frac x a}\right)^2} + 1}\) 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