Derivative of x^5-5x^4+5x^3+1

1. Differentiate $\left(5 x^{3} + \left(x^{5} - 5 x^{4}\right)\right) + 1$ term by term:

   1. Differentiate $5 x^{3} + \left(x^{5} - 5 x^{4}\right)$ term by term:

      1. Differentiate $x^{5} - 5 x^{4}$ term by term:

         1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

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

            1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

            So, the result is: $- 20 x^{3}$

         The result is: $5 x^{4} - 20 x^{3}$

      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: $15 x^{2}$

      The result is: $5 x^{4} - 20 x^{3} + 15 x^{2}$

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

   The result is: $5 x^{4} - 20 x^{3} + 15 x^{2}$

2. Now simplify:

   $$5 x^{2} \left(x^{2} - 4 x + 3\right)$$

The answer is:

$$5 x^{2} \left(x^{2} - 4 x + 3\right)$$