#### Answer

$$f'(x)=4x^3$$
The domain of both $f(x)$ and $f'(x)$ is $R$

#### Work Step by Step

$$f(x)=x^4$$
According to definition, $$f'(x)=\lim\limits_{h\to0}\frac{(x+h)^4-x^4}{h}$$
$$f'(x)=\lim\limits_{h\to0}\frac{[(x+h)^2-x^2][(x+h)^2+x^2]}{h}$$
$$f'(x)=\lim\limits_{h\to0}\frac{(x+h-x)(x+h+x)[(x+h)^2+x^2]}{h}$$
$$f'(x)=\lim\limits_{h\to0}\frac{h(x+h+x)[(x+h)^2+x^2]}{h}$$
$$f'(x)=\lim\limits_{h\to0}(2x+h)[(x+h)^2+x^2]$$
$$f'(x)=2x[x^2+x^2]$$
$$f'(x)=4x^3$$
The domain of both $f(x)$ and $f'(x)$ is $R$