Derivative of 15x⁴+3x³-4x²+5x-1

1. Differentiate $\left(5 x + \left(- 4 x^{2} + \left(15 x^{4} + 3 x^{3}\right)\right)\right) - 1$ term by term:

   1. Differentiate $5 x + \left(- 4 x^{2} + \left(15 x^{4} + 3 x^{3}\right)\right)$ term by term:

      1. Differentiate $- 4 x^{2} + \left(15 x^{4} + 3 x^{3}\right)$ term by term:

         1. Differentiate $15 x^{4} + 3 x^{3}$ 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: $x^{4}$ goes to $4 x^{3}$

               So, the result is: $60 x^{3}$

            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: $9 x^{2}$

            The result is: $60 x^{3} + 9 x^{2}$

         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: $- 8 x$

         The result is: $60 x^{3} + 9 x^{2} - 8 x$

      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: $5$

      The result is: $60 x^{3} + 9 x^{2} - 8 x + 5$

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

   The result is: $60 x^{3} + 9 x^{2} - 8 x + 5$

The answer is:

$$60 x^{3} + 9 x^{2} - 8 x + 5$$