Derivative of Arccosecant Function/Corollary
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Corollary to Derivative of Arccosecant Function
Let $x \in \R$.
Let $\arccsc \dfrac x a$ be the arccosecant of $\dfrac x a$.
Then:
- $\dfrac {\map \d {\arccsc \frac x a} } {\d x} = \dfrac {-a} {\size x {\sqrt {x^2 - a^2} } } = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}$
Proof
| \(\ds \frac {\map \d {\arccsc \frac x a} } {\d x}\) | \(=\) | \(\ds \frac 1 a \frac {-1} {\size {\frac x a} \sqrt {\paren {\frac x a}^2 - 1} }\) | Derivative of Arccosecant Function and Derivative of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-1} {\size {\frac x a} \frac {\sqrt {x^2 - a^2} } a}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-a^2} {\size x {\sqrt {x^2 - a^2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-a} {\size x {\sqrt {x^2 - a^2} } }\) |
$\Box$
Similarly:
| \(\ds \frac {\map \d {\arccsc \frac x a} } {\d x}\) | \(=\) | \(\ds \begin{cases} \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}\) |
Derivative of Arccosecant Function and Derivative of Function of Constant Multiple |
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| \(\ds \) | \(=\) | \(\ds \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
\dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}\) |
simplifying as above |
$\blacksquare$