# Derivative of Inverse Hyperbolic Sine of x over a

## 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 $\displaystyle$ $=$ $\displaystyle \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }$ $\displaystyle$ $=$ $\displaystyle \frac 1 a \frac a {\sqrt {x^2 + a^2} }$ $\displaystyle$ $=$ $\displaystyle \frac 1 {\sqrt {x^2 + a^2} }$

$\blacksquare$