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$(z*e^2iz)\((z+2i)^2)$

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Identical expressions
(z*e^ two iz)\((z+2i)^2)
(z multiply by e squared iz)\((z plus 2i) squared )
(z multiply by e to the power of two iz)\((z plus 2i) squared )
(z*e2iz)\((z+2i)2)
z*e2iz\z+2i2
(z*e²iz)\((z+2i)²)
(z*e to the power of 2iz)\((z+2i) to the power of 2)
(ze^2iz)\((z+2i)^2)
(ze2iz)\((z+2i)2)
ze2iz\z+2i2
ze^2iz\z+2i^2
Similar expressions
(z*e^2iz)\((z-2i)^2)

The derivative of the function/ (z*e^2iz)\((z+2i)^2)

Derivative of (z*e^2iz)\((z+2i)^2)

The solution

You have entered [src]

    2     
 z*E *I*z 
----------
         2
(z + 2*I)

$$\frac{z i e^{2} z}{\left(z + 2 i\right)^{2}}$$

(((z*E^2)*i)*z)/(z + 2*i)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d z} f{\left(z \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $z^{2}$ goes to $2 z$
  So, the result is: $2 i z e^{2}$
To find $\frac{d}{d z} g{\left(z \right)}$ :
1. Let $u = z + 2 i$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d z} \left(z + 2 i\right)$ :
  1. Differentiate $z + 2 i$ term by term:
    1. Apply the power rule: $z$ goes to $1$
    2. The derivative of the constant $2 i$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $2 z + 4 i$
Now plug in to the quotient rule:

$\frac{- i z^{2} \left(2 z + 4 i\right) e^{2} + 2 i z \left(z + 2 i\right)^{2} e^{2}}{\left(z + 2 i\right)^{4}}$
Now simplify:

$- \frac{4 z e^{2}}{z^{3} + 6 i z^{2} - 12 z - 8 i}$

The answer is:

$- \frac{4 z e^{2}}{z^{3} + 6 i z^{2} - 12 z - 8 i}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2          2      2               2
z*E *I + I*z*e    I*z *(-4*I - 2*z)*e 
--------------- + --------------------
            2                   4     
   (z + 2*I)           (z + 2*I)

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

Simplify

The second derivative [src]

    /                    2   \   
    |      4*z        3*z    |  2
2*I*|1 - ------- + ----------|*e 
    |    z + 2*I            2|   
    \              (z + 2*I) /   
---------------------------------
                     2           
            (z + 2*I)

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

Simplify

The third derivative [src]

     /           2             \   
     |        2*z         3*z  |  2
12*I*|-1 - ---------- + -------|*e 
     |              2   z + 2*I|   
     \     (z + 2*I)           /   
-----------------------------------
                      3            
             (z + 2*I)

$$\frac{12 i \left(- \frac{2 z^{2}}{\left(z + 2 i\right)^{2}} + \frac{3 z}{z + 2 i} - 1\right) e^{2}}{\left(z + 2 i\right)^{3}}$$

Simplify

The graph

$Derivative of (z*e^2iz)\((z+2i)^2)$

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How to use it?

Identical expressions

Similar expressions

Derivative of (z*e^2iz)\((z+2i)^2)

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Piecewise:

The solution