By Limit Definition,
#f'(x)=lim_{h to 0}{tan(x+h)-tanx}/h#
by the trig identity: #tan(alpha+beta)={tan alpha +tan beta}/{1-tan alpha tan beta}#,
#=lim_{h to 0}{{tan x+tan h}/{1-tan x tan h}-tan x}/h#
by taking the common denominator,
#=lim_{h to 0}{{tan x + tan h-(tan x - tan^2x tan h)}/{1-tan x tan h}}/h#
by cancelling out #tan x#'s,
#=lim_{h to 0}{{tan h +tan^2x tan h)/{1-tan x tan h}}/h#
by factoring out #tan h#,
#=lim_{h to 0}({tan h}/h cdot {1+tan^2 x}/{1-tan x tan h})#
by #tan h ={sin h}/{cos h}# and #1+tan^2x=sec^2x#,
#=lim_{h to 0}({sin h}/h cdot 1/{cos h} cdot {sec^2x}/{1-tan x tan h})#
by #lim_{h to 0}{sin h}/h=1#,
#=1 cdot 1/1 cdot {sec^2x}/1=sec^2 x#
