Derivative of t^3+3t^2-5t

1. Differentiate $- 5 t + \left(t^{3} + 3 t^{2}\right)$ term by term:

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

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

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

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

         So, the result is: $6 t$

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

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

      1. Apply the power rule: $t$ goes to $1$

      So, the result is: $-5$

   The result is: $3 t^{2} + 6 t - 5$

The answer is:

$$3 t^{2} + 6 t - 5$$