Derivative of Cosecant of Function

Theorem

Let $u$ be a differentiable real function of $x$.

Then:

$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$

where $\sec$ is the cosecant function and $\tan$ is the cotangent function.

$\ds \map {\frac \d {\d x} } {\csc u}$

$=$

$\ds \map {\frac \d {\d u} } {\csc u} \frac {\d u} {\d x}$

$\ds $

$=$

$\ds -\csc u \cot u \frac {\d u} {\d x}$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.19$
1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $8$