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

1. Differentiate $x^{5} - 3 x^{4} - 5 x^{3} + 3 x - 1$ 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. 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: $12 x^{3}$

      So, the result is: $- 12 x^{3}$

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

      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: $15 x^{2}$

      So, the result is: $- 15 x^{2}$

   4. 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: $3$

   5. The derivative of the constant $\left(-1\right) 1$ is zero.

   The result is: $5 x^{4} - 12 x^{3} - 15 x^{2} + 3$

The answer is:

$$5 x^{4} - 12 x^{3} - 15 x^{2} + 3$$