Derivative of f(x)=x^3+3x^2-72x+90

1. Differentiate $\left(- 72 x + \left(x^{3} + 3 x^{2}\right)\right) + 90$ term by term:

   1. Differentiate $- 72 x + \left(x^{3} + 3 x^{2}\right)$ term by term:

      1. Differentiate $x^{3} + 3 x^{2}$ term by term:

         1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

         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: $6 x$

         The result is: $3 x^{2} + 6 x$

      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: $-72$

      The result is: $3 x^{2} + 6 x - 72$

   2. The derivative of the constant $90$ is zero.

   The result is: $3 x^{2} + 6 x - 72$

The answer is:

$$3 x^{2} + 6 x - 72$$