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