Differentiate the function y = cot(x).

The derivative of y = cot(x) is -csc^2(x).

To differentiate y = cot(x), we can use the quotient rule. Let u = 1 and v = sin(x), then:

y = u/v = 1/sin(x)

y' = (v du/dx - u dv/dx) / v^2
= (-1/sin^2(x))(-cos(x)) / sin^2(x)
= cos(x) / sin^3(x)

Recall that cot(x) = 1/tan(x) = cos(x)/sin(x). Therefore, we can simplify y' as:

y' = cos(x) / sin^3(x)
= cos(x) / (sin(x) * sin^2(x))
= cos(x) / (sin(x) * (1 - cos^2(x)))
= cos(x) / (sin(x) - sin(x) * cos^2(x))
= cos(x) / sin(x) * (1 / (1 - cos^2(x)))
= cot(x) * csc^2(x)

Hence, the derivative of y = cot(x) is y' = -csc^2(x).