Derivative of y=4x^4+1/3x^3+1/2x^2+4

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

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

      1. Differentiate $4 x^{4} + \frac{x^{3}}{3}$ 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: $16 x^{3}$

         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: $x^{2}$

         The result is: $16 x^{3} + x^{2}$

      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: $x$

      The result is: $16 x^{3} + x^{2} + x$

   2. The derivative of the constant $4$ is zero.

   The result is: $16 x^{3} + x^{2} + x$

2. Now simplify:

   $$x \left(16 x^{2} + x + 1\right)$$

The answer is:

$$x \left(16 x^{2} + x + 1\right)$$