Derivative of y=sqrt13x^2+5x+8

1. Differentiate $5 x + \left(\sqrt{13 x}\right)^{2} + 8$ term by term:

   1. Let $u = \sqrt{13 x}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{13 x}$:

      1. Let $u = 13 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 13 x$:

         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: $13$

         The result of the chain rule is:

         $$\frac{\sqrt{13}}{2 \sqrt{x}}$$

      The result of the chain rule is:

      $$13$$

   4. 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: $5$

   5. The derivative of the constant $8$ is zero.

   The result is: $18$

The answer is:

$$18$$