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

1 Answer
Sep 19, 2014

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