What is the second derivative of ln(tan(x))?

Answer 1

f'(x) = 1/tan(x) * sec^2(x) where * means multiply and ^ means to the power of.

= cot(x) * sec^2(x)

f''(x) = f'(cot(x)*sec^2(x) + cot(x)*f'[sec^2(x)]

= -csc^2(x)*sec^2(x) + cot(x)*2tan(x)sec^2(x)

= sec^2(x) [cot(x)-csc^2(x)] +2tan(x)cot(x)

= sec^2(x) [cot(x)-csc^2(x)] +2