Derivative of Inverse Hyperbolic Secant of x over a

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Theorem

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

where $0 < x < a$.


Proof

Let $0 < x < a$.

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

\(\displaystyle \frac {\mathrm d \left({\operatorname{sech}^{-1} \frac x a}\right)} {\mathrm d x}\) \(=\) \(\displaystyle \frac 1 a \frac {-1} {\frac x a \sqrt{1 - \left({\frac x a}\right)^2} }\) Derivative of $\operatorname{sech}^{-1}$ and Derivative of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \frac {-a} {x \sqrt{\frac {a^2 - x^2} {a^2} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {-a} {x \sqrt{a^2 - x^2} }\)


$\operatorname{sech}^{-1} \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$.

$\blacksquare$


Also see