Derivative of Real Area Hyperbolic Secant of x over a
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Theorem
- $\dfrac {\map \d {\map \arsech {\frac x a} } } {\d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$
where $0 < x < a$.
Proof
Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
\(\ds \frac {\map \rd {\map \arsech {\frac x a} } } {\rd x}\) | \(=\) | \(\ds \frac 1 a \frac {-1} {\frac x a \sqrt {1 - \paren {\frac x a}^2} }\) | Derivative of $\arsech$ and Derivative of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-a} {x \sqrt {\frac {a^2 - x^2} {a^2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a} {x \sqrt {a^2 - x^2} }\) |
$\arsech \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$.
$\blacksquare$