Derivative of y=xsqrtx-12x+27

1. Differentiate $\sqrt{x} x - 12 x + 27$ 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. 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: $12$

      So, the result is: $-12$

   3. The derivative of the constant $27$ is zero.

   The result is: $\frac{3 \sqrt{x}}{2} - 12$

The answer is:

$$\frac{3 \sqrt{x}}{2} - 12$$