Derivative of y(x)=x^8-3x^3+7x+1

1. Differentiate $\left(7 x + \left(x^{8} - 3 x^{3}\right)\right) + 1$ term by term:

   1. Differentiate $7 x + \left(x^{8} - 3 x^{3}\right)$ term by term:

      1. Differentiate $x^{8} - 3 x^{3}$ term by term:

         1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$

         2. 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: $- 9 x^{2}$

         The result is: $8 x^{7} - 9 x^{2}$

      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: $7$

      The result is: $8 x^{7} - 9 x^{2} + 7$

   2. The derivative of the constant $1$ is zero.

   The result is: $8 x^{7} - 9 x^{2} + 7$

The answer is:

$$8 x^{7} - 9 x^{2} + 7$$