Derivative of Inverse Hyperbolic Cosecant of x over a

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Theorem

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

where $x \ne 0$.


Proof

Let $0 < x < a$.

Then $0 < \dfrac x a < 1$ and so:

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


$\csch^{-1} \dfrac x a$ is not defined when $x = 0$.

$\blacksquare$


Also see