Derivative of sqrt(5x+1)^8

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

2. Apply the power rule: $u^{8}$ goes to $8 u^{7}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{5 x + 1}$:

   1. Let $u = 5 x + 1$.

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

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

      1. Differentiate $5 x + 1$ 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: $5$

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

         The result is: $5$

      The result of the chain rule is:

      $$\frac{5}{2 \sqrt{5 x + 1}}$$

   The result of the chain rule is:

   $$20 \left(5 x + 1\right)^{3}$$

4. Now simplify:

   $$20 \left(5 x + 1\right)^{3}$$

The answer is:

$$20 \left(5 x + 1\right)^{3}$$