Derivative of ln((x+a)^0.5)

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

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

3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \sqrt{a + x}$:

   1. Let $u = a + x$.

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

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$\frac{1}{2 \left(a + x\right)}$$

4. Now simplify:

   $$\frac{1}{2 \left(a + x\right)}$$

The answer is:

$$\frac{1}{2 \left(a + x\right)}$$