Derivative of y=ln(sqrt(5x)-1)

1. Let $u = \sqrt{5 x} - 1$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{5 x} - 1\right)$:

   1. Differentiate $\sqrt{5 x} - 1$ term by term:

      1. Let $u = 5 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

         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: $5$

         The result of the chain rule is:

         $$\frac{\sqrt{5}}{2 \sqrt{x}}$$

      4. The derivative of the constant $-1$ is zero.

      The result is: $\frac{\sqrt{5}}{2 \sqrt{x}}$

   The result of the chain rule is:

   $$\frac{\sqrt{5}}{2 \sqrt{x} \left(\sqrt{5 x} - 1\right)}$$

4. Now simplify:

   $$\frac{\sqrt{5}}{2 \left(- \sqrt{x} + \sqrt{5} x\right)}$$

The answer is: