Let f(x) = tan x

Now, df(x)/dx = lim_h->0 [{f(x + h) - f(x)}/h]

                   = lim_h->0 [{tan (x + h) - tan x}/h]

                   = lim_h->0 [{sin (x + h)/cos (x + h) - sin x/cos x}/h]

                   = lim_h->0 [{sin (x + h) * cos x - cos (x + h) * sin x}/{cos x * cos(x + h) * h}]

                   = lim_h->0 [{sin (x + h - x)/{cos x * cos(x + h) * h}]       {using the formula sin (A + B)}

                   = lim_h->0 [{sin h/{cos x * cos(x + h) * h}]

                   = lim_h->0 (sin h)/h * lim_h->0 {cos x * cos(x + h) ]

                   = 1/(cos x * cos x)       {since lim_h->0 (sin h)/h = 1}

                   = 1/cos² x

                   = sec² x

So, d(tan x)/dx = sec² x