Question

r=log(cos^2(4theta)) what is the derivative of this?

Answer 1

just use the chain rule:

let r = log(u)
dr/dθ = 1/u du/dθ

let u = cos^2(v)
du/dθ = 2cosv(-sinv) dv/dθ

let v = 4θ
dv/dθ = 4

So, dr/dθ = sec^2(4θ) (-2cos4θsin4θ) (4)
= -8tan4θ

Or, you could just recall that
log(cos^2(4θ)) = 2log cos 4θ

so a simpler application of the chain rule would be

2/cos4θ (-4sin4θ)
= -8tan4θ