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\displaystyle [br]\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}x}(f(x) + g(x)) = \frac{\mathrm{d}}{\mathrm{d}x}(f(x)) + \frac{\mathrm{d}}{\mathrm{d}x} (g(x))end{equation*}
\displaystyle [br]\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}x}(\lambda x) = \lambda \frac{\mathrm{d}}{\mathrm{d}x}(x)\end{equation*}
\displaystyle[br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1} \end{equation*}
\displaystyle[br]\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{\mathrm{d}}{\mathrm{d}x}(2x + 4x^2) \\ &= \frac{\mathrm{d}}{\mathrm{d}x}(2x) + \frac{\mathrm{d}}{\mathrm{d}x}(4x^2) \\ & = 2\frac{\mathrm{d}}{\mathrm{d}x}(x) + 4\frac{\mathrm{d}}{\mathrm{d}x}(x^2) \\ & = 2\frac{\mathrm{d}}{\mathrm{d}x}(x^1) + 4\frac{\mathrm{d}}{\mathrm{d}x}(x^2) \\ & = 2(1 \times x^{1-1}) + 4(2 \times x^{2-1}) \\ &= 2(1 \times x^0) + 4(2 \times x) \\ &= 2(1) + 4(2x) \\ & = 2 + 8x \end{align*}