Question

How to derive: f(w)=ln(cos(w-1))

Answer 1

The word you're looking for is "differentiate", not "derive".

$\dfrac{\mathrm d}{\mathrm dw}\ln(\cos(w-1))=\dfrac{\frac{\mathrm d}{\mathrm dw}\cos(w-1)}{\cos(w-1)}=\dfrac{-\sin(w-1)\frac{\mathrm d}{\mathrm dw}(w-1)}{\cos(w-1)}=-\dfrac{\sin(w-1)}{\cos(w-1)}=-\tan(w-1)$