Derivative of y=x^4+13*x^3+53*x^2+73*x+1

1. Differentiate $\left(73 x + \left(53 x^{2} + \left(x^{4} + 13 x^{3}\right)\right)\right) + 1$ term by term:

   1. Differentiate $73 x + \left(53 x^{2} + \left(x^{4} + 13 x^{3}\right)\right)$ term by term:

      1. Differentiate $53 x^{2} + \left(x^{4} + 13 x^{3}\right)$ term by term:

         1. Differentiate $x^{4} + 13 x^{3}$ term by term:

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

            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: $39 x^{2}$

            The result is: $4 x^{3} + 39 x^{2}$

         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: $106 x$

         The result is: $4 x^{3} + 39 x^{2} + 106 x$

      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: $73$

      The result is: $4 x^{3} + 39 x^{2} + 106 x + 73$

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

   The result is: $4 x^{3} + 39 x^{2} + 106 x + 73$

The answer is:

$$4 x^{3} + 39 x^{2} + 106 x + 73$$