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