You entered:

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

What you mean?

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

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

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

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

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

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

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

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

The solution

You have entered [src]

(1 + 3*z)*z*(3 - z)
-------------------
         3

$$\frac{z \left(- z + 3\right) \left(3 z + 1\right)}{3}$$

d /(1 + 3*z)*z*(3 - z)\
--|-------------------|
dz\         3         /

$$\frac{d}{d z} \frac{z \left(- z + 3\right) \left(3 z + 1\right)}{3}$$

Transform

Detail solution

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: $z \left(3 - z\right) - \frac{z \left(3 z + 1\right)}{3} + \frac{\left(3 - z\right) \left(3 z + 1\right)}{3}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

            z*(1 + 3*z)   (1 + 3*z)*(3 - z)
z*(3 - z) - ----------- + -----------------
                 3                3

$$z \left(- z + 3\right) - \frac{z \left(3 z + 1\right)}{3} + \frac{\left(- z + 3\right) \left(3 z + 1\right)}{3}$$

Simplify

The second derivative [src]

2*(8/3 - 3*z)

$$2 \cdot \left(- 3 z + \frac{8}{3}\right)$$

Simplify

The third derivative [src]

-6

$$-6$$

Simplify

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What you mean?

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

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