\[ f(x) = x^2 \]

\begin{align*}
  \frac{d}{dx} f(x) &= \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \\[2ex]
  &= \lim_{\Delta x \to 0} \frac{(x+\Delta x)^2 - x^2}{\Delta x} \\[2ex]
  &= \lim_{\Delta x \to 0} \frac{x^2 + 2x \Delta x + {\Delta x}^2 - x^2}{\Delta x} \\[2ex]
  &= \lim_{\Delta x \to 0} \frac{2x \Delta x + {\Delta x}^2}{\Delta x} \\[2ex]
  &= \lim_{\Delta x \to 0} \frac{2x + \Delta x}{1} \\[2ex]
  &= \lim_{\Delta x \to 0} \frac{2x + \Delta x}{1} \\[2ex]
  &= 2x + 0 \\[2ex]
  &= 2x \\[2ex]
\end{align*}