I need to probe why the derivative of tan (x) is sec^2 (x)?

Question

I need to use the limit:

lim [ f (x+h) - f (x) ] / h
h->0

Answer 1

Consider:-
[ tan(x + h) - tan x ] / h
[(tan x + tan h) / (1 - tan x tan h) - tan x] / h
= N/D
N = (tanx + tan h) - tan x (1 - tan x tan h)
N = tan h + tan²x tan h
N = tan h(1 + tan²x)

D = h (1 - tan x tan h)

N / D = (tan h / h) (1 + tan²x) / (1 - tan x tan h)
N/D = (1) (1 + tan²x) / (1) as h--> 0
N/D = tan²x + 1 = sec²x