(x^4)'=\lim_{h\rightarrow 0}\frac{(x+h)^4-x^4}{h}=\lim_{h\rightarrow 0}\frac{4x^3h + 6x^2h^2+4xh^3+h^4}{h}=\lim_{h\rightarrow 0}(4x^3 + 6x^2h + 4xh^2 + h^3)=4x^3.