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##