Derivative of sqrtx^5+1

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

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

   2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

   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:

      $$\frac{5 x^{\frac{3}{2}}}{2}$$

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

   The result is: $\frac{5 x^{\frac{3}{2}}}{2}$

The answer is:

$$\frac{5 x^{\frac{3}{2}}}{2}$$