Derivative of 2e^(-2x)+√x

1. Differentiate $\sqrt{x} + 2 e^{- 2 x}$ term by term:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let $u = - 2 x$.

      2. The derivative of $e^{u}$ is itself.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 2 x\right)$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $-2$

         The result of the chain rule is:

         $$- 2 e^{- 2 x}$$

      So, the result is: $- 4 e^{- 2 x}$

   2. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

   The result is: $- 4 e^{- 2 x} + \frac{1}{2 \sqrt{x}}$

The answer is:

$$- 4 e^{- 2 x} + \frac{1}{2 \sqrt{x}}$$