Answer
$f(x)=\tan x$ and $a=\frac{\pi}{4}$
Work Step by Step
The derivatif of a function $f$ at point $a$: $f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$
Given: $f'(a)=\lim\limits_{h\to 0}\frac{\tan(\frac{\pi}{4}+h)-1}{h}$
Find $f(x)$ and $a$:
$\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}=\lim\limits_{h\to 0}\frac{\tan(\frac{\pi}{4}+h)-1}{h}$
$\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}=\lim\limits_{h\to 0}\frac{\tan(\frac{\pi}{4}+h)-\tan\frac{\pi}{4}}{h}$
$f(a+h)=\tan(\frac{\pi}{4}+h)$ and $f(a)=\tan\frac{\pi}{4}$
It gives us $f(x)=\tan x$ and $a=\frac{\pi}{4}$.
