find the derivative of f(x)=cot(3x)csc^2(3x)

Answer 1

f(x)=cot(3x)csc^2(3x) = cosx/sin^3x

Let sinx = u . Then du = cosx dx

The integral of f(x) becomes

Integral of du/u^3 = -(1/2) u^-2
= (-1/2)(1/sinx)^2
= (-1/2)csc^2x

Answer 2

first change it to
f(x) = cos 3x/(sin 3x)^3 like drwls had

now use the quotient rule to find f'(x)

f'(x) = [(sin 3x)^3(-3)(sin 3x) - 3(sin 3x)^2(cos 3x)(3)(cos 3x)]/(sin 3x)^6
= -3[(sin 3x)^2 + 3(cos 3x)^2]/(sin 3x)^4

Answer 3

I made at least one mistake. The first was using x instead of 3x.