Derivative of Arccosecant of Function
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Theorem
Let $u$ be a differentiable real function of $x$ such that $\size u > 1$.
Then:
- $\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$
where $\arcsec$ denotes the arccosecant of $x$.
Proof
\(\ds \map {\frac \d {\d x} } {\arccsc u}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {\arccsc u} \frac {\d u} {\d x}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}\) | Derivative of Arccosecant Function |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.25$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $18$