Differentiate the function $\displaystyle y = \sqrt{1 + xe^{-2x}}$

$ \begin{equation} \begin{aligned} \text{if } y = \sqrt{1 + xe^{-2x}} = (1 + xe^{-2x})^{\frac{1}{2}} \text{ then by using Quotient Rule and Product Rule} \\ \\ y' =& \frac{1}{2} (1 + x e ^{-2x})^{\frac{-1}{2}} \cdot [x (e^{-2x} \cdot (-2)) + e^{-2x} (1)] \\ \\ y' =& \frac{e^{-2x} (1 - 2x)}{2 \sqrt{1 + xe^{-2x}}} \end{aligned} \end{equation} $

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