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(z*cos(z))/(1-z^2)

The derivative of the function/ (z*cos(z))/(1+z^2)

Derivative of (z*cos(z))/(1+z^2)

The solution

You have entered [src]

z*cos(z)
--------
      2 
 1 + z

$$\frac{z \cos{\left(z \right)}}{z^{2} + 1}$$

(z*cos(z))/(1 + z^2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(z \right)} = z \cos{\left(z \right)}$ and $g{\left(z \right)} = z^{2} + 1$ .

To find $\frac{d}{d z} f{\left(z \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} = f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$
  
  $f{\left(z \right)} = z$ ; to find $\frac{d}{d z} f{\left(z \right)}$ :
  1. Apply the power rule: $z$ goes to $1$
  $g{\left(z \right)} = \cos{\left(z \right)}$ ; to find $\frac{d}{d z} g{\left(z \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d z} \cos{\left(z \right)} = - \sin{\left(z \right)}$
  The result is: $- z \sin{\left(z \right)} + \cos{\left(z \right)}$
To find $\frac{d}{d z} g{\left(z \right)}$ :
1. Differentiate $z^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $z^{2}$ goes to $2 z$
  The result is: $2 z$
Now plug in to the quotient rule:

$\frac{- 2 z^{2} \cos{\left(z \right)} + \left(z^{2} + 1\right) \left(- z \sin{\left(z \right)} + \cos{\left(z \right)}\right)}{\left(z^{2} + 1\right)^{2}}$
Now simplify:

$- \frac{2 z^{2} \cos{\left(z \right)} + \left(z^{2} + 1\right) \left(z \sin{\left(z \right)} - \cos{\left(z \right)}\right)}{\left(z^{2} + 1\right)^{2}}$

The answer is:

$- \frac{2 z^{2} \cos{\left(z \right)} + \left(z^{2} + 1\right) \left(z \sin{\left(z \right)} - \cos{\left(z \right)}\right)}{\left(z^{2} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                        2       
-z*sin(z) + cos(z)   2*z *cos(z)
------------------ - -----------
           2                  2 
      1 + z           /     2\  
                      \1 + z /

$$- \frac{2 z^{2} \cos{\left(z \right)}}{\left(z^{2} + 1\right)^{2}} + \frac{- z \sin{\left(z \right)} + \cos{\left(z \right)}}{z^{2} + 1}$$

Simplify

The second derivative [src]

                                                      /         2 \       
                                                      |      4*z  |       
                                                  2*z*|-1 + ------|*cos(z)
                                                      |          2|       
                       4*z*(-cos(z) + z*sin(z))       \     1 + z /       
-2*sin(z) - z*cos(z) + ------------------------ + ------------------------
                                     2                          2         
                                1 + z                      1 + z          
--------------------------------------------------------------------------
                                       2                                  
                                  1 + z

$$\frac{- z \cos{\left(z \right)} + \frac{4 z \left(z \sin{\left(z \right)} - \cos{\left(z \right)}\right)}{z^{2} + 1} + \frac{2 z \left(\frac{4 z^{2}}{z^{2} + 1} - 1\right) \cos{\left(z \right)}}{z^{2} + 1} - 2 \sin{\left(z \right)}}{z^{2} + 1}$$

Simplify

The third derivative [src]

                         /         2 \                                                          /         2 \       
                         |      4*z  |                                                        2 |      2*z  |       
                       6*|-1 + ------|*(-cos(z) + z*sin(z))                               24*z *|-1 + ------|*cos(z)
                         |          2|                                                          |          2|       
                         \     1 + z /                        6*z*(2*sin(z) + z*cos(z))         \     1 + z /       
-3*cos(z) + z*sin(z) - ------------------------------------ + ------------------------- - --------------------------
                                           2                                 2                            2         
                                      1 + z                             1 + z                     /     2\          
                                                                                                  \1 + z /          
--------------------------------------------------------------------------------------------------------------------
                                                            2                                                       
                                                       1 + z

$$\frac{- \frac{24 z^{2} \left(\frac{2 z^{2}}{z^{2} + 1} - 1\right) \cos{\left(z \right)}}{\left(z^{2} + 1\right)^{2}} + z \sin{\left(z \right)} + \frac{6 z \left(z \cos{\left(z \right)} + 2 \sin{\left(z \right)}\right)}{z^{2} + 1} - 3 \cos{\left(z \right)} - \frac{6 \left(z \sin{\left(z \right)} - \cos{\left(z \right)}\right) \left(\frac{4 z^{2}}{z^{2} + 1} - 1\right)}{z^{2} + 1}}{z^{2} + 1}$$

Simplify

The graph

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Derivative of (z*cos(z))/(1+z^2)

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Piecewise:

The solution