Demostración derivada tg x

y = tg x

y = (sen x)/ (cos x)

ln y = ln [(sen x)/ (cos x)] = ln (sen x) – ln (cos x)

y’/ y = x’ [(cos x)/ (sen x)] – x’ [(-sen x)/ (cos x)]

y’/ y = x’ [(cos x)/ (sen x)] + x’ [(sen x)/ (cos x)]

y’/ y = x’ [(cos²x + sen²x)/ (sen x · cos x)]

y’ = yx’ [(cos²x + sen²x)/ (sen x · cos x)]

y’ = x’ [(sen x)/ (cos x)] [(cos²x + sen²x)/ (sen x · cos x)]

y’ = x’ [(cos²x + sen²x)/ (cos²x)]

y’ = x’ (1 + tg²x) = x’ + x’ tg²x