Derivative of f(x)=5x^3+6x^2-7x-8

1. Differentiate $\left(- 7 x + \left(5 x^{3} + 6 x^{2}\right)\right) - 8$ term by term:

   1. Differentiate $- 7 x + \left(5 x^{3} + 6 x^{2}\right)$ term by term:

      1. Differentiate $5 x^{3} + 6 x^{2}$ 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: $15 x^{2}$

         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: $12 x$

         The result is: $15 x^{2} + 12 x$

      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: $-7$

      The result is: $15 x^{2} + 12 x - 7$

   2. The derivative of the constant $-8$ is zero.

   The result is: $15 x^{2} + 12 x - 7$

The answer is:

$$15 x^{2} + 12 x - 7$$