Derivative of sqrt(x)*3-2*x/x^(3/2)

1. Differentiate $\sqrt{x} 3 - \frac{2 x}{x^{\frac{3}{2}}}$ 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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

      So, the result is: $\frac{3}{2 \sqrt{x}}$

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

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

         1. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = x$ and $g{\left(x \right)} = x^{\frac{3}{2}}$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

            1. Apply the power rule: $x$ goes to $1$

            To find $\frac{d}{d x} g{\left(x \right)}$:

            1. Apply the power rule: $x^{\frac{3}{2}}$ goes to $\frac{3 \sqrt{x}}{2}$

            Now plug in to the quotient rule:

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

         So, the result is: $- \frac{1}{x^{\frac{3}{2}}}$

      So, the result is: $\frac{1}{x^{\frac{3}{2}}}$

   The result is: $\frac{3}{2 \sqrt{x}} + \frac{1}{x^{\frac{3}{2}}}$

2. Now simplify:

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

The answer is:

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