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Find the derivative of f(x) = x2 + 2x using first principle of derivative.

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+2\mathrm{x} \\ \Rightarrow \mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)={\left(\mathrm{x}+\mathrm{h}\right)}^2+2\left(\mathrm{x}+\mathrm{h}\right)={\mathrm{x}}^2+{\mathrm{h}}^2+2\mathrm{xh}+2\mathrm{x}+2\mathrm{h} \\ \mathrm{By} \mathrm{first} \mathrm{principal} \mathrm{we} \mathrm{have}: \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\underset{\mathrm{h}\to 0}{\lim } \frac{\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}} \\ =\underset{\mathrm{h}\to 0}{\lim } \frac{\left({\mathrm{x}}^2+{\mathrm{h}}^2+2\mathrm{xh}+2\mathrm{x}+2\mathrm{h}\right)-\left({\mathrm{x}}^2+2\mathrm{x}\right)}{\mathrm{h}} \\ =\underset{\mathrm{h}\to 0}{\lim } \frac{{\mathrm{x}}^2+{\mathrm{h}}^2+2\mathrm{xh}+2\mathrm{x}+2\mathrm{h}-{\mathrm{x}}^2-2\mathrm{x}}{\mathrm{h}} \\ =\underset{\mathrm{h}\to 0}{\lim } \frac{{\mathrm{h}}^2+2\mathrm{xh}+2\mathrm{h}}{\mathrm{h}} \\ =\underset{\mathrm{h}\to 0}{\lim } \frac{\mathrm{h}\left(\mathrm{h}+2\mathrm{x}+2\right)}{\mathrm{h}} \\ =\underset{\mathrm{h}\to 0}{\lim } \left(\mathrm{h}+2\mathrm{x}+2\right) \\ \mathrm{Applying} \mathrm{limit} \mathrm{we} \mathrm{get}: \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\left(0+2\mathrm{x}+2\right)=2\mathrm{x}+2 \left[\mathrm{Answer}\right] \end{gathered}$$
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