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How do I us the Limit definition of derivative on ##f(x)=tan(x)##?
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##