Derivative of ax+b/x^(1/2)

1. Differentiate $a x + \frac{b}{\sqrt{x}}$ 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: $a$

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

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

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

      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:

         $$- \frac{1}{2 x^{\frac{3}{2}}}$$

      So, the result is: $- \frac{b}{2 x^{\frac{3}{2}}}$

   The result is: $a - \frac{b}{2 x^{\frac{3}{2}}}$

The answer is:

$$a - \frac{b}{2 x^{\frac{3}{2}}}$$