Derivative of ln*(arccos*1/sqrt(x))

1. Let $u = \frac{\operatorname{acos}{\left(1 \right)}}{\sqrt{x}}$.

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

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

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

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

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

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

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

         The result of the chain rule is:

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

      So, the result is: $- \frac{\operatorname{acos}{\left(1 \right)}}{2 x^{\frac{3}{2}}}$

   The result of the chain rule is:

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

The answer is:

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