The derivative of cosec x is -cot x cosec x.
Cosec x is the reciprocal of sin x, so we can rewrite it as (1/sin x). Using the quotient rule of differentiation, we can find the derivative of cosec x as follows:
d/dx (cosec x) = d/dx (1/sin x) = (-1/sin^2x) * d/dx (sin x)
Now, using the chain rule, we can find the derivative of sin x:
d/dx (sin x) = cos x
Substituting this back into our original expression, we get:
d/dx (cosec x) = (-1/sin^2x) * cos x
Simplifying this, we get:
d/dx (cosec x) = -cot x cosec x