The derivative of the cosecant function, $\cosec x$, can be derived using the quotient rule or by recognizing it as the reciprocal of the sine function. Differentiating $\cosec x$ results in $-\cosec x \cot x$, which can be rewritten in terms of cosine and cosecant. This leads to the expression $-\cos x \cosec x$, confirming the relationship between the functions involved.