Find the derivative of f(x) (csc(2x^2+1)^4

Answer 1

just use the chain rule:

f = u^4
u = csc(v)
v = 2x^2+1

f' = 4u^3 u'
u' = -csc(v) cot(v) v'
v' = 4x

So,

f' = 4 (csc(2x^2+1)^3 * (-csc(2x^2+1)*cot(2x^2+1)) * (4x)
= -16x csc^4(2x^2+1) cot(2x^2+1)