\[\begin{aligned} \frac{d}{dx}(x^{2}) &= \frac{dx^{2}}{dx} \\ &= \frac{f(x+dx)-f(x)}{dx} \\ &= \frac{(x+dx)^{2}-x^{2}}{dx} \\ &= \frac{x^{2}+2xdx+dx^{2}-x^{2}}{dx} \\ &= \frac{2xdx+dx^{2}}{dx} \\ &= 2x+dx \\ &= 2x \text{} \end{aligned} \]
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