Derivative of y=x^4-2x^3+5x^2-x+12

1. Differentiate $x^{4} - 2 x^{3} + 5 x^{2} - x + 12$ term by term:

   1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

   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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $6 x^{2}$

      So, the result is: $- 6 x^{2}$

   3. 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: $10 x$

   4. 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: $-1$

   5. The derivative of the constant $12$ is zero.

   The result is: $4 x^{3} - 6 x^{2} + 10 x - 1$

The answer is:

$$4 x^{3} - 6 x^{2} + 10 x - 1$$