Derivative of ((1+3z)/3*z)(3-z)

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

   1. Apply the product rule:

      $$\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} h{\left(z \right)} = f{\left(z \right)} g{\left(z \right)} \frac{d}{d z} h{\left(z \right)} + f{\left(z \right)} h{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} h{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$$

      $f{\left(z \right)} = z$; to find $\frac{d}{d z} f{\left(z \right)}$:

      1. Apply the power rule: $z$ goes to $1$

      $g{\left(z \right)} = 3 z + 1$; to find $\frac{d}{d z} g{\left(z \right)}$:

      1. Differentiate $3 z + 1$ term by term:

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

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

            1. Apply the power rule: $z$ goes to $1$

            So, the result is: $3$

         The result is: $3$

      $h{\left(z \right)} = 3 - z$; to find $\frac{d}{d z} h{\left(z \right)}$:

      1. Differentiate $3 - z$ term by term:

         1. The derivative of the constant $3$ is zero.

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

            1. Apply the power rule: $z$ goes to $1$

            So, the result is: $-1$

         The result is: $-1$

      The result is: $3 z \left(3 - z\right) - z \left(3 z + 1\right) + \left(3 - z\right) \left(3 z + 1\right)$

   So, the result is: $z \left(3 - z\right) - \frac{z \left(3 z + 1\right)}{3} + \frac{\left(3 - z\right) \left(3 z + 1\right)}{3}$

2. Now simplify:

   $$- 3 z^{2} + \frac{16 z}{3} + 1$$

The answer is:

$$- 3 z^{2} + \frac{16 z}{3} + 1$$