Derivative of x^12+4x^17+x

1. Differentiate $x + \left(4 x^{17} + x^{12}\right)$ term by term:

   1. Differentiate $4 x^{17} + x^{12}$ term by term:

      1. Apply the power rule: $x^{12}$ goes to $12 x^{11}$

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

         1. Apply the power rule: $x^{17}$ goes to $17 x^{16}$

         So, the result is: $68 x^{16}$

      The result is: $68 x^{16} + 12 x^{11}$

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

   The result is: $68 x^{16} + 12 x^{11} + 1$

The answer is:

$$68 x^{16} + 12 x^{11} + 1$$