Derivative of y=xsqrt(x)-15x+72

1. Differentiate $\left(\sqrt{x} x - 15 x\right) + 72$ term by term:

   1. Differentiate $\sqrt{x} x - 15 x$ term by term:

      1. Apply the product rule:

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

         $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

         $g{\left(x \right)} = \sqrt{x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

      2. 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: $-15$

      The result is: $\frac{3 \sqrt{x}}{2} - 15$

   2. The derivative of the constant $72$ is zero.

   The result is: $\frac{3 \sqrt{x}}{2} - 15$

The answer is:

$$\frac{3 \sqrt{x}}{2} - 15$$