Derivative of 3x^7-6x^6-4x^3+5x^2+17

1. Differentiate $3 x^{7} - 6 x^{6} - 4 x^{3} + 5 x^{2} + 17$ 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: $x^{7}$ goes to $7 x^{6}$

      So, the result is: $21 x^{6}$

   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: $x^{6}$ goes to $6 x^{5}$

         So, the result is: $36 x^{5}$

      So, the result is: $- 36 x^{5}$

   3. 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: $12 x^{2}$

      So, the result is: $- 12 x^{2}$

   4. 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: $10 x$

   5. The derivative of the constant $17$ is zero.

   The result is: $21 x^{6} - 36 x^{5} - 12 x^{2} + 10 x$

2. Now simplify:

   $$x \left(21 x^{5} - 36 x^{4} - 12 x + 10\right)$$

The answer is:

$$x \left(21 x^{5} - 36 x^{4} - 12 x + 10\right)$$