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Identical expressions
((z^ two)(z-3i)^ two)/(((z^ two)+ nine)^ two)
((z squared )(z minus 3i) squared ) divide by (((z squared ) plus 9) squared )
((z to the power of two)(z minus 3i) to the power of two) divide by (((z to the power of two) plus nine) to the power of two)
((z2)(z-3i)2)/(((z2)+9)2)
z2z-3i2/z2+92
((z²)(z-3i)²)/(((z²)+9)²)
((z to the power of 2)(z-3i) to the power of 2)/(((z to the power of 2)+9) to the power of 2)
z^2z-3i^2/z^2+9^2
((z^2)(z-3i)^2) divide by (((z^2)+9)^2)
Similar expressions
((z^2)(z+3i)^2)/(((z^2)+9)^2)
((z^2)(z-3i)^2)/(((z^2)-9)^2)

The derivative of the function/ ((z^2)(z-3i)^2)/(((z^2)+9)^2)

Derivative of ((z^2)(z-3i)^2)/(((z^2)+9)^2)

The solution

You have entered [src]

 2          2
z *(z - 3*I) 
-------------
          2  
  / 2    \   
  \z  + 9/

$$\frac{z^{2} \left(z - 3 i\right)^{2}}{\left(z^{2} + 9\right)^{2}}$$

(z^2*(z - 3*i)^2)/(z^2 + 9)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d z} \frac{f{\left(z \right)}}{g{\left(z \right)}} = \frac{- f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}}{g^{2}{\left(z \right)}}$

$f{\left(z \right)} = z^{2} \left(z - 3 i\right)^{2}$ and $g{\left(z \right)} = \left(z^{2} + 9\right)^{2}$ .

To find $\frac{d}{d z} f{\left(z \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} = f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$
  
  $f{\left(z \right)} = z^{2}$ ; to find $\frac{d}{d z} f{\left(z \right)}$ :
  1. Apply the power rule: $z^{2}$ goes to $2 z$
  $g{\left(z \right)} = \left(z - 3 i\right)^{2}$ ; to find $\frac{d}{d z} g{\left(z \right)}$ :
  1. Let $u = z - 3 i$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d z} \left(z - 3 i\right)$ :
    1. Differentiate $z - 3 i$ term by term:
      1. Apply the power rule: $z$ goes to $1$
      2. The derivative of the constant $- 3 i$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $2 z - 6 i$
  The result is: $z^{2} \left(2 z - 6 i\right) + 2 z \left(z - 3 i\right)^{2}$
To find $\frac{d}{d z} g{\left(z \right)}$ :
1. Let $u = z^{2} + 9$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d z} \left(z^{2} + 9\right)$ :
  1. Differentiate $z^{2} + 9$ term by term:
    1. The derivative of the constant $9$ is zero.
    2. Apply the power rule: $z^{2}$ goes to $2 z$
    The result is: $2 z$
  The result of the chain rule is:
  
  $2 z \left(2 z^{2} + 18\right)$
Now plug in to the quotient rule:

$\frac{- 2 z^{3} \left(z - 3 i\right)^{2} \left(2 z^{2} + 18\right) + \left(z^{2} + 9\right)^{2} \left(z^{2} \left(2 z - 6 i\right) + 2 z \left(z - 3 i\right)^{2}\right)}{\left(z^{2} + 9\right)^{4}}$
Now simplify:

$\frac{6 z \left(i z^{3} + 9 z^{2} - 27 i z - 27\right)}{z^{6} + 27 z^{4} + 243 z^{2} + 729}$

The answer is:

$\frac{6 z \left(i z^{3} + 9 z^{2} - 27 i z - 27\right)}{z^{6} + 27 z^{4} + 243 z^{2} + 729}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 2                             2      3          2
z *(-6*I + 2*z) + 2*z*(z - 3*I)    4*z *(z - 3*I) 
-------------------------------- - ---------------
                   2                          3   
           / 2    \                   / 2    \    
           \z  + 9/                   \z  + 9/

$$- \frac{4 z^{3} \left(z - 3 i\right)^{2}}{\left(z^{2} + 9\right)^{3}} + \frac{z^{2} \left(2 z - 6 i\right) + 2 z \left(z - 3 i\right)^{2}}{\left(z^{2} + 9\right)^{2}}$$

Simplify

The second derivative [src]

  /                                                                                /         2 \\
  |                                                                   2          2 |      6*z  ||
  |                                                                2*z *(z - 3*I) *|-1 + ------||
  |                                     2                                          |          2||
  | 2            2                   8*z *(z - 3*I)*(-3*I + 2*z)                   \     9 + z /|
2*|z  + (z - 3*I)  + 4*z*(z - 3*I) - --------------------------- + -----------------------------|
  |                                                  2                              2           |
  \                                             9 + z                          9 + z            /
-------------------------------------------------------------------------------------------------
                                                    2                                            
                                            /     2\                                             
                                            \9 + z /

$$\frac{2 \left(\frac{2 z^{2} \left(z - 3 i\right)^{2} \left(\frac{6 z^{2}}{z^{2} + 9} - 1\right)}{z^{2} + 9} - \frac{8 z^{2} \left(z - 3 i\right) \left(2 z - 3 i\right)}{z^{2} + 9} + z^{2} + 4 z \left(z - 3 i\right) + \left(z - 3 i\right)^{2}\right)}{\left(z^{2} + 9\right)^{2}}$$

Simplify

The third derivative [src]

   /                                                                     /         2 \       /         2 \                       \
   |                                                        3          2 |      8*z  |       |      6*z  |                       |
   |                                                     2*z *(z - 3*I) *|-3 + ------|   2*z*|-1 + ------|*(z - 3*I)*(-3*I + 2*z)|
   |                 / 2            2                \                   |          2|       |          2|                       |
   |             2*z*\z  + (z - 3*I)  + 4*z*(z - 3*I)/                   \     9 + z /       \     9 + z /                       |
12*|-3*I + 2*z - ------------------------------------- - ----------------------------- + ----------------------------------------|
   |                                  2                                    2                                   2                 |
   |                             9 + z                             /     2\                               9 + z                  |
   \                                                               \9 + z /                                                      /
----------------------------------------------------------------------------------------------------------------------------------
                                                                    2                                                             
                                                            /     2\                                                              
                                                            \9 + z /

$$\frac{12 \left(- \frac{2 z^{3} \left(z - 3 i\right)^{2} \left(\frac{8 z^{2}}{z^{2} + 9} - 3\right)}{\left(z^{2} + 9\right)^{2}} + \frac{2 z \left(z - 3 i\right) \left(2 z - 3 i\right) \left(\frac{6 z^{2}}{z^{2} + 9} - 1\right)}{z^{2} + 9} + 2 z - \frac{2 z \left(z^{2} + 4 z \left(z - 3 i\right) + \left(z - 3 i\right)^{2}\right)}{z^{2} + 9} - 3 i\right)}{\left(z^{2} + 9\right)^{2}}$$

Simplify

The graph

Derivative of ((z^2)(z-3i)^2)/(((z^2)+9)^2)

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Identical expressions

Similar expressions

Derivative of ((z^2)(z-3i)^2)/(((z^2)+9)^2)

The graph:

Piecewise:

The solution