Derivada de x/((x+1)(x^2+1))

Ha introducido [src]

       x        
----------------
        / 2    \
(x + 1)*\x  + 1/

\frac{x}{\left(x + 1\right) \left(x^{2} + 1\right)}

x/(((x + 1)*(x^2 + 1)))

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x$ y $g{\left(x \right)} = \left(x + 1\right) \left(x^{2} + 1\right)$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x + 1$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. diferenciamos $x + 1$ miembro por miembro:
    1. La derivada de una constante $1$ es igual a cero.
    2. Según el principio, aplicamos: $x$ tenemos $1$
    Como resultado de: $1$
  $g{\left(x \right)} = x^{2} + 1$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $x^{2} + 1$ miembro por miembro:
    1. La derivada de una constante $1$ es igual a cero.
    2. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
    Como resultado de: $2 x$
  Como resultado de: $x^{2} + 2 x \left(x + 1\right) + 1$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- x \left(x^{2} + 2 x \left(x + 1\right) + 1\right) + \left(x + 1\right) \left(x^{2} + 1\right)}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}}$

Respuesta:

$\frac{- x \left(x^{2} + 2 x \left(x + 1\right) + 1\right) + \left(x + 1\right) \left(x^{2} + 1\right)}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                     /      2              \
       1           x*\-1 - x  - 2*x*(x + 1)/
---------------- + -------------------------
        / 2    \                        2   
(x + 1)*\x  + 1/              2 / 2    \    
                       (x + 1) *\x  + 1/

\frac{x \left(- x^{2} - 2 x \left(x + 1\right) - 1\right)}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)}

Simplificar

Segunda derivada [src]

              /                2                                                               /     2              \\              
        2     |           1 + x  + 2*x*(1 + x)   /  1      2*x  \ /     2              \   2*x*\1 + x  + 2*x*(1 + x)/|              
-2 - 2*x  + x*|-2 - 6*x + -------------------- + |----- + ------|*\1 + x  + 2*x*(1 + x)/ + --------------------------| - 4*x*(1 + x)
              |                  1 + x           |1 + x        2|                                         2          |              
              \                                  \        1 + x /                                    1 + x           /              
------------------------------------------------------------------------------------------------------------------------------------
                                                                          2                                                         
                                                                2 /     2\                                                          
                                                         (1 + x) *\1 + x /

\frac{- 2 x^{2} - 4 x \left(x + 1\right) + x \left(- 6 x + \frac{2 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} + \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right) - 2 + \frac{x^{2} + 2 x \left(x + 1\right) + 1}{x + 1}\right) - 2}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}}

Simplificar

4-я производная [src]

  /        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          /                          2                     \                                                                                                                 /                          2                     \                                                                                                                                                          \                                                                                                                                                                                                                                                                                                                                                          \
  |        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                   /     2              \ |   1         1         4*x            2*x       |                                                                                          /     2              \ |   1         1         4*x            2*x       |                                                                                                                                                          |                                                                                                                                                                                                                                                                                                                                                          |
  |        |                                                                                                                                                                                                                                                                   /  1      2*x  \ /     2              \   /  1      2*x  \ /     2              \                                                                                          /  1      2*x  \   2*\1 + x  + 2*x*(1 + x)/*|-------- - ------ + --------- + ----------------|                                                                   /  1      2*x  \   4*x*\1 + x  + 2*x*(1 + x)/*|-------- - ------ + --------- + ----------------|      2 /  1      2*x  \ /     2              \                                                                    /  1      2*x  \ /     2              \|                                                                                                                                                                                                                      /  1      2*x  \ /     2              \                                                      /  1      2*x  \ /     2              \|
  |        |                                                                                                                                                                                                                                                                   |----- + ------|*\1 + x  + 2*x*(1 + x)/   |----- + ------|*\1 + x  + 2*x*(1 + x)/                                                                              2*(1 + 3*x)*|----- + ------|                            |       2        2           2           /     2\|                                                     4*x*(1 + 3*x)*|----- + ------|                              |       2        2           2           /     2\|   4*x *|----- + ------|*\1 + x  + 2*x*(1 + x)/                                                                2*x*|----- + ------|*\1 + x  + 2*x*(1 + x)/|                                                                                                                                                                                                                    2*|----- + ------|*\1 + x  + 2*x*(1 + x)/                                                  4*x*|----- + ------|*\1 + x  + 2*x*(1 + x)/|
  |        |                                   /                          2                     \                            /                                                3                                   2      \     /     2              \                          |1 + x        2|                          |1 + x        2|                              2                  /     2              \     /     2              \               |1 + x        2|                            |(1 + x)    1 + x    /     2\    (1 + x)*\1 + x /|       3 /     2              \                                    |1 + x        2|                              |(1 + x)    1 + x    /     2\    (1 + x)*\1 + x /|        |1 + x        2|                               /     2              \       2 /     2              \       |1 + x        2|                       |     /     2              \                            /                          2                     \     /     2              \                                                     2 /     2              \     |1 + x        2|                                                /     2              \       |1 + x        2|                       |
  |        |  18    12*(1 + 3*x)               |   1         1         4*x            2*x       |     /     2              \ |   1              1              4*x         8*x             2*x                 4*x       |   6*\1 + x  + 2*x*(1 + x)/   8*(1 + 3*x)    36*x    \        1 + x /                          \        1 + x /                          48*x *(1 + 3*x)   18*x*\1 + x  + 2*x*(1 + x)/   6*\1 + x  + 2*x*(1 + x)/               \        1 + x /                            \                    \1 + x /                    /   48*x *\1 + x  + 2*x*(1 + x)/    32*x*(1 + 3*x)                  \        1 + x /                              \                    \1 + x /                    /        \        1 + x /                          18*x*\1 + x  + 2*x*(1 + x)/   36*x *\1 + x  + 2*x*(1 + x)/       \        1 + x /                       |   6*\1 + x  + 2*x*(1 + x)/     /     2              \ |   1         1         4*x            2*x       |   4*\1 + x  + 2*x*(1 + x)/               /  1      2*x  \   12*(1 + 3*x)   24*x *\1 + x  + 2*x*(1 + x)/     \        1 + x /                          24*x*(1 + 3*x)   16*x*\1 + x  + 2*x*(1 + x)/       \        1 + x /                       |
2*|-12 + x*|----- - ------------ - 4*(1 + 3*x)*|-------- - ------ + --------- + ----------------| + 3*\1 + x  + 2*x*(1 + x)/*|-------- - ---------------- - --------- + --------- + ----------------- + -----------------| + ------------------------ + ----------- + ------ + --------------------------------------- - --------------------------------------- - --------------- - --------------------------- - ------------------------ - ---------------------------- + --------------------------------------------------------------------------- + ---------------------------- - ---------------- - ------------------------------ + ----------------------------------------------------------------------------- + -------------------------------------------- + --------------------------- + ---------------------------- + -------------------------------------------| - ------------------------ - 4*\1 + x  + 2*x*(1 + x)/*|-------- - ------ + --------- + ----------------| + ------------------------ + 4*(1 + 3*x)*|----- + ------| + ------------ - ---------------------------- - ----------------------------------------- + -------------- - --------------------------- - -------------------------------------------|
  |        |1 + x            2                 |       2        2           2           /     2\|                            |       3           /     2\           2           3          2 /     2\                   2|                  3                   2          2                          2                                        2                              2                       2                        /     2\                  1 + x                                                  1 + x                                                       3                     /     2\                    2                                                        2                                                               2                                 2 /     2\                             2                               /     2\             |                  2                                    |       2        2           2           /     2\|                 2                        |1 + x        2|      1 + x                        2                               1 + x                              2                     /     2\                                2                  |
  |        |          (1 + x)                  |(1 + x)    1 + x    /     2\    (1 + x)*\1 + x /|                            |(1 + x)    (1 + x)*\1 + x /   /     2\    /     2\    (1 + x) *\1 + x /           /     2\ |           (1 + x)               1 + x      1 + x                    (1 + x)                                    1 + x                       /     2\                /     2\                 (1 + x)*\1 + x /                                                                                                                             /     2\              (1 + x)*\1 + x /               1 + x                                                    1 + x                                                        /     2\                           (1 + x) *\1 + x /                     /     2\                        (1 + x)*\1 + x /             |           (1 + x)                                     |(1 + x)    1 + x    /     2\    (1 + x)*\1 + x /|            1 + x                         \        1 + x /                           /     2\                                                              1 + x              (1 + x)*\1 + x /                           1 + x                   |
  \        \                                   \                    \1 + x /                    /                            \                              \1 + x /    \1 + x /                        (1 + x)*\1 + x / /                                                                                                                                            \1 + x /                \1 + x /                                                                                                                                                              \1 + x /                                                                                                                                                                   \1 + x /                                                         (1 + x)*\1 + x /                                                     /                                                       \                    \1 + x /                    /                                                                                     \1 + x /                                                                                                                                                    /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       2 /     2\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (1 + x) *\1 + x /

\frac{2 \left(- \frac{24 x^{2} \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{24 x \left(3 x + 1\right)}{x^{2} + 1} + x \left(\frac{48 x^{3} \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x^{2} + 1\right)^{3}} - \frac{48 x^{2} \left(3 x + 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{4 x^{2} \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{36 x^{2} \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)^{2}} - \frac{4 x \left(3 x + 1\right) \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right)}{x^{2} + 1} + \frac{4 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right) \left(\frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{1}{x^{2} + 1} + \frac{1}{\left(x + 1\right)^{2}}\right)}{x^{2} + 1} + \frac{36 x}{x^{2} + 1} - \frac{18 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{32 x \left(3 x + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)} + \frac{2 x \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)} + \frac{18 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)} - 4 \left(3 x + 1\right) \left(\frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{1}{x^{2} + 1} + \frac{1}{\left(x + 1\right)^{2}}\right) + \frac{8 \left(3 x + 1\right)}{x^{2} + 1} + 3 \left(x^{2} + 2 x \left(x + 1\right) + 1\right) \left(\frac{8 x^{3}}{\left(x^{2} + 1\right)^{3}} + \frac{4 x^{2}}{\left(x + 1\right) \left(x^{2} + 1\right)^{2}} - \frac{4 x}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)} - \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)} + \frac{1}{\left(x + 1\right)^{3}}\right) - \frac{\left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} - \frac{2 \left(3 x + 1\right) \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right)}{x + 1} + \frac{2 \left(x^{2} + 2 x \left(x + 1\right) + 1\right) \left(\frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{1}{x^{2} + 1} + \frac{1}{\left(x + 1\right)^{2}}\right)}{x + 1} + \frac{18}{x + 1} - \frac{6 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{12 \left(3 x + 1\right)}{\left(x + 1\right)^{2}} + \frac{\left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right)^{2}} + \frac{6 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right)^{3}}\right) - \frac{4 x \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} - \frac{16 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)} + 4 \left(3 x + 1\right) \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) - 4 \left(x^{2} + 2 x \left(x + 1\right) + 1\right) \left(\frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{1}{x^{2} + 1} + \frac{1}{\left(x + 1\right)^{2}}\right) - 12 + \frac{4 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} + \frac{12 \left(3 x + 1\right)}{x + 1} - \frac{2 \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x + 1} - \frac{6 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right)^{2}}\right)}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}}

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Tercera derivada [src]

              /                                                                                                                                                                                     /  1      2*x  \ /     2              \                                                       /  1      2*x  \ /     2              \                             \                                                                                                    
              |                                                                                                                                                                                     |----- + ------|*\1 + x  + 2*x*(1 + x)/                                                   2*x*|----- + ------|*\1 + x  + 2*x*(1 + x)/                             |                                                                                                    
              |                    /     2              \                                                           /                          2                     \     /     2              \   |1 + x        2|                                               2 /     2              \       |1 + x        2|                              /     2              \|     /     2              \                                                   /     2              \
              |    6*(1 + 3*x)   2*\1 + x  + 2*x*(1 + x)/               /  1      2*x  \     /     2              \ |   1         1         4*x            2*x       |   3*\1 + x  + 2*x*(1 + x)/   \        1 + x /                          12*x*(1 + 3*x)   12*x *\1 + x  + 2*x*(1 + x)/       \        1 + x /                          8*x*\1 + x  + 2*x*(1 + x)/|   3*\1 + x  + 2*x*(1 + x)/     /  1      2*x  \ /     2              \   6*x*\1 + x  + 2*x*(1 + x)/
-6 - 18*x - x*|6 - ----------- - ------------------------ - 2*(1 + 3*x)*|----- + ------| + 2*\1 + x  + 2*x*(1 + x)/*|-------- - ------ + --------- + ----------------| + ------------------------ + --------------------------------------- - -------------- + ---------------------------- + ------------------------------------------- + --------------------------| + ------------------------ + 3*|----- + ------|*\1 + x  + 2*x*(1 + x)/ + --------------------------
              |       1 + x                    2                        |1 + x        2|                            |       2        2           2           /     2\|                  2                            1 + x                             2                        2                                     2                                  /     2\     |            1 + x               |1 + x        2|                                         2          
              |                           1 + x                         \        1 + x /                            |(1 + x)    1 + x    /     2\    (1 + x)*\1 + x /|           (1 + x)                                                          1 + x                 /     2\                                 1 + x                           (1 + x)*\1 + x /     |                                \        1 + x /                                    1 + x           
              \                                                                                                     \                    \1 + x /                    /                                                                                                  \1 + x /                                                                                      /                                                                                                    
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                                                                                                                                                                                                                             (1 + x) *\1 + x /

\frac{- x \left(\frac{12 x^{2} \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{12 x \left(3 x + 1\right)}{x^{2} + 1} + \frac{2 x \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} + \frac{8 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)} - 2 \left(3 x + 1\right) \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) + 2 \left(x^{2} + 2 x \left(x + 1\right) + 1\right) \left(\frac{4 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{\left(x + 1\right) \left(x^{2} + 1\right)} - \frac{1}{x^{2} + 1} + \frac{1}{\left(x + 1\right)^{2}}\right) + 6 - \frac{2 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} - \frac{6 \left(3 x + 1\right)}{x + 1} + \frac{\left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x + 1} + \frac{3 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{\left(x + 1\right)^{2}}\right) - 18 x + \frac{6 x \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x^{2} + 1} + 3 \left(\frac{2 x}{x^{2} + 1} + \frac{1}{x + 1}\right) \left(x^{2} + 2 x \left(x + 1\right) + 1\right) - 6 + \frac{3 \left(x^{2} + 2 x \left(x + 1\right) + 1\right)}{x + 1}}{\left(x + 1\right)^{2} \left(x^{2} + 1\right)^{2}}

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Derivada de x/((x+1)(x^2+1))

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