Differential Quotient

Using the $h$-method we can write $$f'(x) = \lim_{h\rightarrow 0} \frac{(x+h)^3-x^3}{h}$$ Expanding the first term in the denominator gives $$f'(x) = \lim_{h\rightarrow 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3-x^3}{h}$$ After canceling out all $h$ we get $$f'(x) = \lim_{h\rightarrow 0} 3x^2 + 3xh^2 + h^3$$ And finally with calculating the limit $h\rightarrow 0$ we obtain the final result $$f'(x) = 3x^2$$ as the derivative of the function $f(x) = x^3$