The derivative of csc(x) with respect to x is -cot(x)csc(x). One can derive the derivative of the cosecant function, csc(x), by using the chain rule.
The chain rule of differentiation states that the derivative of y = f(g(x)) with respect to x is dy/dx = dy/dt dt/dx, where t = g(x). The definition of the cosecant function is csc(x) = 1/sin(x). Let y = csc(x) and t = sin(x). In this case, dy/dt = -1/(sinx)^2 and dt/dx = cos(x). Therefore, dy/dx = -cos(x)/(sin(x))^2. One can substitute the definitions csc(x) = 1/sin(x) and cot(x) = 1/tan(x) = cos(x)/sin(x) into this result to derive d(csc(x))/dx = -cot(x)csc(x).