Derivative of sqrt(e^(3x+1))

1. Let $u = e^{3 x + 1}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{3 x + 1}$:

   1. Let $u = 3 x + 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 1\right)$:

      1. Differentiate $3 x + 1$ term by term:

         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: $3$

         2. The derivative of the constant $1$ is zero.

         The result is: $3$

      The result of the chain rule is:

      $$3 e^{3 x + 1}$$

   The result of the chain rule is:

   $$\frac{3 e^{- \frac{3 x}{2} - \frac{1}{2}} e^{3 x + 1}}{2}$$

4. Now simplify:

   $$\frac{3 e^{\frac{3 x}{2} + \frac{1}{2}}}{2}$$

The answer is:

$$\frac{3 e^{\frac{3 x}{2} + \frac{1}{2}}}{2}$$