Mister Exam

You entered:

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

What you mean?

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
(1 + 3*z)*z*(3 - z)
-------------------
         3         
z(z+3)(3z+1)3\frac{z \left(- z + 3\right) \left(3 z + 1\right)}{3}
d /(1 + 3*z)*z*(3 - z)\
--|-------------------|
dz\         3         /
ddzz(z+3)(3z+1)3\frac{d}{d z} \frac{z \left(- z + 3\right) \left(3 z + 1\right)}{3}
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Apply the product rule:

      ddzf(z)g(z)h(z)=f(z)g(z)ddzh(z)+f(z)h(z)ddzg(z)+g(z)h(z)ddzf(z)\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(z)=zf{\left(z \right)} = z; to find ddzf(z)\frac{d}{d z} f{\left(z \right)}:

      1. Apply the power rule: zz goes to 11

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

      1. Differentiate 3z+13 z + 1 term by term:

        1. The derivative of the constant 11 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: zz goes to 11

          So, the result is: 33

        The result is: 33

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

      1. Differentiate 3z3 - z term by term:

        1. The derivative of the constant 33 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: zz goes to 11

          So, the result is: 1-1

        The result is: 1-1

      The result is: 3z(3z)z(3z+1)+(3z)(3z+1)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(3z)z(3z+1)3+(3z)(3z+1)3z \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:

    3z2+16z3+1- 3 z^{2} + \frac{16 z}{3} + 1


The answer is:

3z2+16z3+1- 3 z^{2} + \frac{16 z}{3} + 1

The graph
02468-8-6-4-2-1010-20002000
The first derivative [src]
            z*(1 + 3*z)   (1 + 3*z)*(3 - z)
z*(3 - z) - ----------- + -----------------
                 3                3        
z(z+3)z(3z+1)3+(z+3)(3z+1)3z \left(- z + 3\right) - \frac{z \left(3 z + 1\right)}{3} + \frac{\left(- z + 3\right) \left(3 z + 1\right)}{3}
The second derivative [src]
2*(8/3 - 3*z)
2(3z+83)2 \cdot \left(- 3 z + \frac{8}{3}\right)
The third derivative [src]
-6
6-6
The graph
Derivative of ((1+3z)/3*z)(3-z)