Derivative of Cosecant Function

Theorem

 * $\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$

where $\sin x \ne 0$.

Proof
From the definition of the cosecant function:
 * $\csc x = \dfrac 1 {\sin x}$

From Derivative of Sine Function:
 * $\map {\dfrac \d {\d x} } {\sin x} = \cos x$

Then:

This is valid only when $\sin x \ne 0$.

Also see

 * Derivative of Sine Function
 * Derivative of Cosine Function


 * Derivative of Tangent Function
 * Derivative of Cotangent Function


 * Derivative of Secant Function