Question

find the derivative of tan^ 3 x using first principle

Answer 1

tan^3(x+h) - tan^3(x) = -1/8 sinh sec^3x (3cos(4x+2h)+cos2h-4) sec^3(x+h)

in the limit that is (since lim sinh/h = 1)
-1/8 sec^6x*3(-8sin^2x cos^2x)
3sec^4x sin^2x
3tan^2x sec^2x
per the chain rule