Derivative of y=-x³+9x²+x-1

1. Differentiate $\left(x + \left(- x^{3} + 9 x^{2}\right)\right) - 1$ term by term:

   1. Differentiate $x + \left(- x^{3} + 9 x^{2}\right)$ term by term:

      1. Differentiate $- x^{3} + 9 x^{2}$ 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: $x^{3}$ goes to $3 x^{2}$

            So, the result is: $- 3 x^{2}$

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

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

            So, the result is: $18 x$

         The result is: $- 3 x^{2} + 18 x$

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

      The result is: $- 3 x^{2} + 18 x + 1$

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

   The result is: $- 3 x^{2} + 18 x + 1$

The answer is:

$$- 3 x^{2} + 18 x + 1$$