Derivada de x*log(x+sqrt(x^2+2))

Ha introducido [src]

     /       ________\
     |      /  2     |
x*log\x + \/  x  + 2 /

x \log{\left(x + \sqrt{x^{2} + 2} \right)}

x*log(x + sqrt(x^2 + 2))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
$g{\left(x \right)} = \log{\left(x + \sqrt{x^{2} + 2} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = x + \sqrt{x^{2} + 2}$ .
2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x + \sqrt{x^{2} + 2}\right)$ :
  1. diferenciamos $x + \sqrt{x^{2} + 2}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. Sustituimos $u = x^{2} + 2$ .
    3. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x^{2} + 2\right)$ :
      1. diferenciamos $x^{2} + 2$ miembro por miembro:
        
        Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
        
        La derivada de una constante $2$ es igual a cero.
        
        Como resultado de: $2 x$
      Como resultado de la secuencia de reglas:
      
      $\frac{x}{\sqrt{x^{2} + 2}}$
    Como resultado de: $\frac{x}{\sqrt{x^{2} + 2}} + 1$
  Como resultado de la secuencia de reglas:
  
  $\frac{\frac{x}{\sqrt{x^{2} + 2}} + 1}{x + \sqrt{x^{2} + 2}}$
Como resultado de: $\frac{x \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)}{x + \sqrt{x^{2} + 2}} + \log{\left(x + \sqrt{x^{2} + 2} \right)}$
Simplificamos:

$\frac{x}{\sqrt{x^{2} + 2}} + \log{\left(x + \sqrt{x^{2} + 2} \right)}$

Respuesta:

$\frac{x}{\sqrt{x^{2} + 2}} + \log{\left(x + \sqrt{x^{2} + 2} \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

  /         x     \                       
x*|1 + -----------|                       
  |       ________|                       
  |      /  2     |      /       ________\
  \    \/  x  + 2 /      |      /  2     |
------------------- + log\x + \/  x  + 2 /
         ________                         
        /  2                              
  x + \/  x  + 2

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

Simplificar

Segunda derivada [src]

      /                 2              \              
      |/         x     \            2  |              
      ||1 + -----------|           x   |              
      ||       ________|    -1 + ------|              
      ||      /      2 |              2|              
      |\    \/  2 + x  /         2 + x |       2*x    
2 - x*|------------------ + -----------| + -----------
      |        ________        ________|      ________
      |       /      2        /      2 |     /      2 
      \ x + \/  2 + x       \/  2 + x  /   \/  2 + x  
------------------------------------------------------
                          ________                    
                         /      2                     
                   x + \/  2 + x

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

Simplificar

Tercera derivada [src]

  /                   3                                           /        2  \\                      2                  
  |  /         x     \        /        2  \     /         x     \ |       x   ||     /         x     \      /        2  \
  |2*|1 + -----------|        |       x   |   3*|1 + -----------|*|-1 + ------||   3*|1 + -----------|      |       x   |
  |  |       ________|    3*x*|-1 + ------|     |       ________| |          2||     |       ________|    3*|-1 + ------|
  |  |      /      2 |        |          2|     |      /      2 | \     2 + x /|     |      /      2 |      |          2|
  |  \    \/  2 + x  /        \     2 + x /     \    \/  2 + x  /              |     \    \/  2 + x  /      \     2 + x /
x*|-------------------- + ----------------- + ---------------------------------| - -------------------- - ---------------
  |                  2               3/2           ________ /       ________\  |            ________           ________  
  | /       ________\        /     2\             /      2  |      /      2 |  |           /      2           /      2   
  | |      /      2 |        \2 + x /           \/  2 + x  *\x + \/  2 + x  /  |     x + \/  2 + x          \/  2 + x    
  \ \x + \/  2 + x  /                                                          /                                         
-------------------------------------------------------------------------------------------------------------------------
                                                            ________                                                     
                                                           /      2                                                      
                                                     x + \/  2 + x

\frac{x \left(\frac{3 x \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right) \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x + \sqrt{x^{2} + 2}\right) \sqrt{x^{2} + 2}} + \frac{2 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{3}}{\left(x + \sqrt{x^{2} + 2}\right)^{2}}\right) - \frac{3 \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\sqrt{x^{2} + 2}} - \frac{3 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{2}}{x + \sqrt{x^{2} + 2}}}{x + \sqrt{x^{2} + 2}}

Simplificar

6-я производная [src]

  /      /                                                                                                                                                                                                                                                              2                                                                                                                                                    2                  \                                                                                                                                                         2                                                                                                  \
  |      |  /           4         2          6  \                      6                         3            /        2  \ /        2          4  \                         2                          2 /        2          4  \                       2 /        2  \                        4 /        2  \                         /        2          4  \                         3 /        2  \        /        2  \                   |                       5        /        2          4  \                        /        2          4  \                         2          /        2  \                                           3 /        2  \                          2 /        2  \|
  |      |  |       35*x      15*x       21*x   |     /         x     \             /        2  \             |       x   | |     6*x        5*x   |            /        2  \          /         x     \  |     6*x        5*x   |      /         x     \  |       x   |       /         x     \  |       x   |       /         x     \ |    10*x        7*x   |        /         x     \  |       x   |        |       x   |  /         x     \|      /         x     \         |    10*x        7*x   |      /         x     \ |     6*x        5*x   |            /        2  \           |       x   |  /         x     \       /         x     \  |       x   |         /         x     \  |       x   ||
  |      |3*|-1 - --------- + ------ + ---------|   8*|1 + -----------|             |       x   |           3*|-1 + ------|*|1 - ------ + ---------|          2 |       x   |        6*|1 + -----------| *|1 - ------ + ---------|   18*|1 + -----------| *|-1 + ------|    24*|1 + -----------| *|-1 + ------|   6*x*|1 + -----------|*|3 - ------ + ---------|   24*x*|1 + -----------| *|-1 + ------|   24*x*|-1 + ------| *|1 + -----------||   48*|1 + -----------|    30*x*|3 - ------ + ---------|   30*|1 + -----------|*|1 - ------ + ---------|            |       x   |        60*|-1 + ------| *|1 + -----------|   120*|1 + -----------| *|-1 + ------|   120*x*|1 + -----------| *|-1 + ------||
  |      |  |             2        2           3|     |       ________|           2*|-1 + ------|             |          2| |         2           2|       6*x *|-1 + ------|          |       ________|  |         2           2|      |       ________|  |          2|       |       ________|  |          2|       |       ________| |         2           2|        |       ________|  |          2|        |          2|  |       ________||      |       ________|         |         2           2|      |       ________| |         2           2|       60*x*|-1 + ------|           |          2|  |       ________|       |       ________|  |          2|         |       ________|  |          2||
  |      |  |     /     2\    2 + x    /     2\ |     |      /      2 |             |          2|             \     2 + x / |    2 + x    /     2\ |            |          2|          |      /      2 |  |    2 + x    /     2\ |      |      /      2 |  \     2 + x /       |      /      2 |  \     2 + x /       |      /      2 | |    2 + x    /     2\ |        |      /      2 |  \     2 + x /        \     2 + x /  |      /      2 ||      |      /      2 |         |    2 + x    /     2\ |      |      /      2 | |    2 + x    /     2\ |            |          2|           \     2 + x /  |      /      2 |       |      /      2 |  \     2 + x /         |      /      2 |  \     2 + x /|
  |      |  \     \2 + x /             \2 + x / /     \    \/  2 + x  /             \     2 + x /                           \             \2 + x / /            \     2 + x /          \    \/  2 + x  /  \             \2 + x / /      \    \/  2 + x  /                      \    \/  2 + x  /                      \    \/  2 + x  / \             \2 + x / /        \    \/  2 + x  /                                      \    \/  2 + x  /|      \    \/  2 + x  /         \             \2 + x / /      \    \/  2 + x  / \             \2 + x / /            \     2 + x /                          \    \/  2 + x  /       \    \/  2 + x  /                        \    \/  2 + x  /               |
3*|- 5*x*|--------------------------------------- + -------------------- + ------------------------------ + ---------------------------------------- + --------------------------- + --------------------------------------------- + ------------------------------------ + ----------------------------------- + ---------------------------------------------- + ------------------------------------- + -------------------------------------| + --------------------- + ----------------------------- + --------------------------------------------- + --------------------------- + ----------------------------------- + ------------------------------------ + --------------------------------------|
  |      |                      5/2                                   5                                 2                 2 /       ________\                  3 /       ________\                                        2                                        3                                        4                     5/2 /       ________\                                             3                                      2    |                      4                     5/2                            3/2 /       ________\                   2 /       ________\                                 2                                       3                                       2    |
  |      |              /     2\                     /       ________\             3/2 /       ________\          /     2\  |      /      2 |          /     2\  |      /      2 |                   3/2 /       ________\                        /       ________\               ________ /       ________\              /     2\    |      /      2 |                        3/2 /       ________\                    2 /       ________\     |     /       ________\              /     2\                       /     2\    |      /      2 |           /     2\  |      /      2 |                /       ________\              ________ /       ________\                   3/2 /       ________\     |
  |      |              \2 + x /                     |      /      2 |     /     2\    |      /      2 |          \2 + x / *\x + \/  2 + x  /          \2 + x / *\x + \/  2 + x  /           /     2\    |      /      2 |               /     2\ |      /      2 |              /      2  |      /      2 |              \2 + x /   *\x + \/  2 + x  /                /     2\    |      /      2 |            /     2\  |      /      2 |     |     |      /      2 |              \2 + x /                       \2 + x /   *\x + \/  2 + x  /           \2 + x / *\x + \/  2 + x  /       /     2\ |      /      2 |             /      2  |      /      2 |           /     2\    |      /      2 |     |
  \      \                                           \x + \/  2 + x  /     \2 + x /   *\x + \/  2 + x  /                                                                                     \2 + x /   *\x + \/  2 + x  /               \2 + x /*\x + \/  2 + x  /            \/  2 + x  *\x + \/  2 + x  /                                                           \2 + x /   *\x + \/  2 + x  /            \2 + x / *\x + \/  2 + x  /     /     \x + \/  2 + x  /                                                                                                                       \2 + x /*\x + \/  2 + x  /           \/  2 + x  *\x + \/  2 + x  /           \2 + x /   *\x + \/  2 + x  /     /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                      ________                                                                                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                                                                                                     /      2                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                               x + \/  2 + x

\frac{3 \left(- 5 x \left(\frac{6 x^{2} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{2}}{\left(x + \sqrt{x^{2} + 2}\right) \left(x^{2} + 2\right)^{3}} + \frac{6 x \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right) \left(\frac{7 x^{4}}{\left(x^{2} + 2\right)^{2}} - \frac{10 x^{2}}{x^{2} + 2} + 3\right)}{\left(x + \sqrt{x^{2} + 2}\right) \left(x^{2} + 2\right)^{\frac{5}{2}}} + \frac{24 x \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right) \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{2}}{\left(x + \sqrt{x^{2} + 2}\right)^{2} \left(x^{2} + 2\right)^{2}} + \frac{24 x \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{3} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x + \sqrt{x^{2} + 2}\right)^{3} \left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{21 x^{6}}{\left(x^{2} + 2\right)^{3}} - \frac{35 x^{4}}{\left(x^{2} + 2\right)^{2}} + \frac{15 x^{2}}{x^{2} + 2} - 1\right)}{\left(x^{2} + 2\right)^{\frac{5}{2}}} + \frac{3 \left(\frac{x^{2}}{x^{2} + 2} - 1\right) \left(\frac{5 x^{4}}{\left(x^{2} + 2\right)^{2}} - \frac{6 x^{2}}{x^{2} + 2} + 1\right)}{\left(x + \sqrt{x^{2} + 2}\right) \left(x^{2} + 2\right)^{2}} + \frac{6 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{2} \left(\frac{5 x^{4}}{\left(x^{2} + 2\right)^{2}} - \frac{6 x^{2}}{x^{2} + 2} + 1\right)}{\left(x + \sqrt{x^{2} + 2}\right)^{2} \left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{2 \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{3}}{\left(x + \sqrt{x^{2} + 2}\right)^{2} \left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{18 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{2} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{2}}{\left(x + \sqrt{x^{2} + 2}\right)^{3} \left(x^{2} + 2\right)} + \frac{24 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{4} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x + \sqrt{x^{2} + 2}\right)^{4} \sqrt{x^{2} + 2}} + \frac{8 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{6}}{\left(x + \sqrt{x^{2} + 2}\right)^{5}}\right) + \frac{30 x \left(\frac{7 x^{4}}{\left(x^{2} + 2\right)^{2}} - \frac{10 x^{2}}{x^{2} + 2} + 3\right)}{\left(x^{2} + 2\right)^{\frac{5}{2}}} + \frac{60 x \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{2}}{\left(x + \sqrt{x^{2} + 2}\right) \left(x^{2} + 2\right)^{2}} + \frac{120 x \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{2} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x + \sqrt{x^{2} + 2}\right)^{2} \left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{30 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right) \left(\frac{5 x^{4}}{\left(x^{2} + 2\right)^{2}} - \frac{6 x^{2}}{x^{2} + 2} + 1\right)}{\left(x + \sqrt{x^{2} + 2}\right) \left(x^{2} + 2\right)^{\frac{3}{2}}} + \frac{60 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right) \left(\frac{x^{2}}{x^{2} + 2} - 1\right)^{2}}{\left(x + \sqrt{x^{2} + 2}\right)^{2} \left(x^{2} + 2\right)} + \frac{120 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{3} \left(\frac{x^{2}}{x^{2} + 2} - 1\right)}{\left(x + \sqrt{x^{2} + 2}\right)^{3} \sqrt{x^{2} + 2}} + \frac{48 \left(\frac{x}{\sqrt{x^{2} + 2}} + 1\right)^{5}}{\left(x + \sqrt{x^{2} + 2}\right)^{4}}\right)}{x + \sqrt{x^{2} + 2}}

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Gráfico

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Derivada de x*log(x+sqrt(x^2+2))

Gráfico:

Definida a trozos:

Solución