Derivative of px-(0,5x^2+2x+6)

1. Differentiate $p x + \left(\left(- \frac{x^{2}}{2} - 2 x\right) - 6\right)$ 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: $p$

   2. Differentiate $\left(- \frac{x^{2}}{2} - 2 x\right) - 6$ term by term:

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

            So, the result is: $- x$

         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: $-2$

         The result is: $- x - 2$

      2. The derivative of the constant $-6$ is zero.

      The result is: $- x - 2$

   The result is: $p - x - 2$

The answer is:

$$p - x - 2$$