Derivative of 8sqrt(5x+1/5x-1)

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

   1. Let $u = \left(\frac{x}{5} + 5 x\right) - 1$.

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

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

      1. Differentiate $\left(\frac{x}{5} + 5 x\right) - 1$ term by term:

         1. Differentiate $\frac{x}{5} + 5 x$ 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: $5$

            2. 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: $\frac{1}{5}$

            The result is: $\frac{26}{5}$

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

         The result is: $\frac{26}{5}$

      The result of the chain rule is:

      $$\frac{13}{5 \sqrt{\left(\frac{x}{5} + 5 x\right) - 1}}$$

   So, the result is: $\frac{104}{5 \sqrt{\left(\frac{x}{5} + 5 x\right) - 1}}$

2. Now simplify:

   $$\frac{104 \sqrt{5}}{5 \sqrt{26 x - 5}}$$

The answer is: