(x^5)'=\lim_{h\rightarrow 0}\frac{(x+h)^5-x^5}{h}=\lim_{h\rightarrow 0}\frac{5x^4h + 10x^3h^2+10x^2h^3+5xh^4 + h^5}{h}=\lim_{h\rightarrow 0}(5x^4 + 10x^3h + 10x^2h^2 + 5xh^3 + h^4)=5x^4.