Simplify using Tan(A + B) identity.

Question

Simplify:

f' (x) = \(\lim\limits_{h\to0} \frac{tan(x+h) - tan x}{h}\)

Answer 1

f'(x) = \(\lim\limits_{h\to0} \frac {tan (x + h)- tan\, x}{h}\)

\(\lim\limits_{h\to0} \frac {tan\,x + tan\,h}{\frac{1-tan\,x\, tan\,- tanx} {h}}\)

(∴ tan (x + h) = \(\frac {tan\,x + tan\,h}{1-tan\,x\, tan\,h} \))

= \(\lim\limits_{h\to0} \frac {tan\,x+tan\,h-tan\,x+ tan^2x\,tan\,h}{h}\)

= \(\lim\limits_{h\to0} \frac {tan\,h (1+tan^2x)}{h}\)

= \(\lim\limits_{h\to0} \frac {tan\,h }{h} \)x (1 + tan²x)

= 1 + tan²x (∴\(\lim\limits_{h\to0} \frac {tan\,h }{h} \)= 1)

= Sec²x  (∴1 + tan²x = sec²x)