Derivative of Arccosecant Function
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Theorem
Let $\operatorname{arccsc} x$ be the arccosecant of $x$.
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Then:
- $D_x \left({\operatorname{arccsc} x}\right) = \dfrac {-1} { \left|{x}\right| \sqrt {x^2 - 1} }$
Proof
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle D_x \operatorname{arccsc} x\) | \(=\) | \(\displaystyle D_x \left( \dfrac \pi 2 - \operatorname{arcsec} x \right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Arcsecant and Arccosecant | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {-1} { \left \vert {x}\right \vert \sqrt {x^2 - 1} }\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Derivative of Constant, Linear Combination of Derivatives, Derivative of Arcsecant Function |
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Sources
- Weisstein, Eric W. "Inverse Cosecant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosecant.html
- Weisstein, Eric W. "Inverse Secant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSecant.html
