derivative of (4Cos^2(3x)+3)^2

Answer 1

Multiply it out and differentiate it one term at a time. Multiplication results in
16 cos^4(3x) + 24 cos^2(3x) + 9

The derivative of the last term is zero.

Let u = cos (3x) and use the function of a function "chain rule" for the other two terms

The derivative of 24 cos^2(3x) (or 24 u^2) is
48 u du/dx = 48 cos(3x)*(-sin(3x))*3
= -144 sin(3x)cos(3x) = -72 sin(6x)

I leave the other term up to you.