Question

Find the derivative of f(x) = tan x at x = 0

Answer 1

We have

f(x) = tan x

$\because f^{\prime }\left(a\right)=li{m}_{h\to 0}\frac{f(a+h)-f\left(a\right)}{h}$

$\because f^{\prime }\left(0\right)=li{m}_{h\to 0}\frac{f(0+h)-f\left(0\right)}{h}$

$=li{m}_{h\to 0}\frac{tanh-tan0}{h}$

$=li{m}_{h\to 0}\frac{tanh}{h}$

$=1[\because li{m}_{\theta \to 0}\frac{tan\theta }{\theta }=1]$

$\therefore f^{\prime }\left(0\right)=1$