Derivative of ln((2x+1)^(1/2))

1. Let $u = \sqrt{2 x + 1}$.

2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x + 1}$:

   1. Let $u = 2 x + 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 1\right)$:

      1. Differentiate $2 x + 1$ 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: $2$

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

         The result is: $2$

      The result of the chain rule is:

      $$\frac{1}{\sqrt{2 x + 1}}$$

   The result of the chain rule is:

   $$\frac{1}{2 x + 1}$$

4. Now simplify:

   $$\frac{1}{2 x + 1}$$

The answer is: