#### Answer

$f(x)=x^6$ and $a=2$

#### Work Step by Step

According to definition, $$f'(x)=\lim\limits_{h\to0}\frac{f(x+h)-f(x)}{h}$$
Therefore, at a number a, the derivative would be $$f'(a)=\lim\limits_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Now we look at the given formula: $$f'(a)=\lim\limits_{h\to0}\frac{(2+h)^6-64}{h}$$
$$f'(a)=\lim\limits_{h\to0}\frac{(2+h)^6-2^6}{h}$$
Here, from the definition, we can deduce that $a=2$.
That means $f(a)=2^6$. Therefore, $f(x)=x^6$