Saved Bookmarks
| 1. |
Simplify:f' (x) = \(\lim\limits_{h\to0} \frac{tan(x+h) - tan x}{h}\) |
|
Answer» 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) |
|