Derivative of Cosecant Function

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Theorem

$D_x \left({\csc x}\right) = -\csc x \cot x$

where $\sin x \ne 0$.


Proof

From the definition of the cosecant function:

$\csc x = \dfrac 1 {\sin x}$

From Derivative of Sine Function:

$D_x \left({\sin x}\right) = \cos x$


Then:

\(\displaystyle D_x \left({\csc x}\right)\) \(=\) \(\displaystyle \cos x \frac {-1} {\sin^2 x}\) Chain Rule for Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1} {\sin x} \frac {\cos x} {\sin x}\)
\(\displaystyle \) \(=\) \(\displaystyle -\csc x \cot x\) Definitions of secant and cotangent

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

$\blacksquare$


Sources