Derivada de y=(lnx)/((x²+4)²sin³x)

Ha introducido [src]

      log(x)     
-----------------
        2        
/ 2    \     3   
\x  + 4/ *sin (x)

\frac{\log{\left(x \right)}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}

log(x)/(((x^2 + 4)^2*sin(x)^3))

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = \log{\left(x \right)}$ y $g{\left(x \right)} = \left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \left(x^{2} + 4\right)^{2}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = x^{2} + 4$ .
  2. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x^{2} + 4\right)$ :
    1. diferenciamos $x^{2} + 4$ miembro por miembro:
      1. La derivada de una constante $4$ es igual a cero.
      2. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
      Como resultado de: $2 x$
    Como resultado de la secuencia de reglas:
    
    $2 x \left(2 x^{2} + 8\right)$
  $g{\left(x \right)} = \sin^{3}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \sin{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  Como resultado de: $2 x \left(2 x^{2} + 8\right) \sin^{3}{\left(x \right)} + 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- \left(2 x \left(2 x^{2} + 8\right) \sin^{3}{\left(x \right)} + 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}{x}}{\left(x^{2} + 4\right)^{4} \sin^{6}{\left(x \right)}}$
Simplificamos:

$\frac{- x \left(4 x \sin{\left(x \right)} + \left(3 x^{2} + 12\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \left(x^{2} + 4\right) \sin{\left(x \right)}}{x \left(x^{2} + 4\right)^{3} \sin^{4}{\left(x \right)}}$

Respuesta:

$\frac{- x \left(4 x \sin{\left(x \right)} + \left(3 x^{2} + 12\right) \cos{\left(x \right)}\right) \log{\left(x \right)} + \left(x^{2} + 4\right) \sin{\left(x \right)}}{x \left(x^{2} + 4\right)^{3} \sin^{4}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/        1        \                                                               
|-----------------|                                                               
|        2        |   /                                   2               \       
|/ 2    \     3   |   |         3    / 2    \     / 2    \     2          |       
\\x  + 4/ *sin (x)/   \- 4*x*sin (x)*\x  + 4/ - 3*\x  + 4/ *sin (x)*cos(x)/*log(x)
------------------- + ------------------------------------------------------------
         x                                         4                              
                                           / 2    \     6                         
                                           \x  + 4/ *sin (x)

\frac{\left(- 4 x \left(x^{2} + 4\right) \sin^{3}{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right)^{4} \sin^{6}{\left(x \right)}} + \frac{\frac{1}{\left(x^{2} + 4\right)^{2}} \frac{1}{\sin^{3}{\left(x \right)}}}{x}

Simplificar

Segunda derivada [src]

       /                                                                   2                                          2                                                                                                                                        \                                            
       |                                                           /     2\     2           2    /     2\     /     2\     2         2    2           /     2\                   /  /     2\                    \              /  /     2\                    \|                                            
       |/3*cos(x)    4*x  \ /  /     2\                    \   - 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)   3*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x)   4*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/|                                            
       ||-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/ - ------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + ------------------------------------|*log(x)                                     
       || sin(x)         2|                                                                                   /     2\                                                                           sin(x)                                        2               |            /  /     2\                    \
  1    \\           4 + x /                                                                                   \4 + x /*sin(x)                                                                                                             4 + x                /          2*\3*\4 + x /*cos(x) + 4*x*sin(x)/
- -- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------
   2                                                                                                                           /     2\                                                                                                                                             /     2\                
  x                                                                                                                            \4 + x /*sin(x)                                                                                                                                    x*\4 + x /*sin(x)         
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                     2                                                                                                                                                      
                                                                                                                                             /     2\     3                                                                                                                                                 
                                                                                                                                             \4 + x / *sin (x)

\frac{\frac{\left(\frac{4 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{2 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x \left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{1}{x^{2}}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}

Simplificar

Tercera derivada [src]

     /                                                                                                                                                                                                                                                                                                                                                                                                                                                             /            2                                          2                                                     \                                                                                                                                                                                                                                                                                                                                                                                                                            \                                                                                                                                                                                                                                                                                                          
     |                                                                                                                                                                                                                                                                                                                                                                                                                                         /3*cos(x)    4*x  \ |    /     2\     2           2    /     2\     /     2\     2         2    2           /     2\              |                                                                                                                                                                                                                                                       /3*cos(x)    4*x  \ /  /     2\                    \              /3*cos(x)    4*x  \ /  /     2\                    \                                               |                                                                                                                                                                                                                                                                                                          
     |                                                                                                                                                                    /          2                                                           2                                                                                                  \                                                                                          |-------- + ------|*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/        /            2                                          2                                                     \     /            2                                          2                                                     \          3*|-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x)   4*x*|-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/                                               |                                                                                                                                                                                                                                                                                                          
     |                                 /                   2            2                    \     /  /     2\                    \                                       |  /     2\     3             3              3    /     2\     /     2\     2                   2    /     2\              2    2                     2    /     2\       |         2    /  /     2\                    \       2 /  /     2\                    \   | sin(x)         2|                                                                                                                        |    /     2\     2           2    /     2\     /     2\     2         2    2           /     2\              |     |    /     2\     2           2    /     2\     /     2\     2         2    2           /     2\              |            | sin(x)         2|                                               | sin(x)         2|                                         /  /     2\                    \       |            /                                                                   2                                          2                                                                                                                                        \                                     
     |/  /     2\                    \ |      4      12*cos (x)     24*x        24*x*cos(x)  |   4*\3*\4 + x /*cos(x) + 4*x*sin(x)/     /     2\                        3*\2*\4 + x / *cos (x) + 8*x*sin (x) - 12*x*sin (x)*\4 + x / - 7*\4 + x / *sin (x)*cos(x) + 12*sin (x)*\4 + x /*cos(x) + 24*x *sin (x)*cos(x) + 24*x*cos (x)*\4 + x /*sin(x)/   21*cos (x)*\3*\4 + x /*cos(x) + 4*x*sin(x)/   40*x *\3*\4 + x /*cos(x) + 4*x*sin(x)/   \           4 + x /                                                                                                                   12*x*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/   9*\- 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)/*cos(x)     \           4 + x /                                               \           4 + x /                                    48*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x)|            |                                                           /     2\     2           2    /     2\     /     2\     2         2    2           /     2\                   /  /     2\                    \              /  /     2\                    \|                                     
     |\3*\4 + x /*cos(x) + 4*x*sin(x)/*|3 - ------ + ---------- + --------- + ---------------| - ---------------------------------- + 9*\4 + x /*cos(x) + 12*x*sin(x) + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------- + -------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------- + -------------------------------------------------------- + --------------------------------------------|*log(x)     |/3*cos(x)    4*x  \ /  /     2\                    \   - 3*\4 + x / *sin (x) + 4*sin (x)*\4 + x / + 6*\4 + x / *cos (x) + 8*x *sin (x) + 24*x*\4 + x /*cos(x)*sin(x)   3*\3*\4 + x /*cos(x) + 4*x*sin(x)/*cos(x)   4*x*\3*\4 + x /*cos(x) + 4*x*sin(x)/|                                     
     |                                 |         2       2                2   /     2\       |                      2                                                                                                                                  /     2\    2                                                                                                         2                                              2                                                                            /     2\                                                                                                                              2                                                                                                                /     2\    2                                                                                      sin(x)                                                            2                                          /     2\                      |          3*||-------- + ------|*\3*\4 + x /*cos(x) + 4*x*sin(x)/ - ------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + ------------------------------------|                                     
     |                                 |    4 + x     sin (x)     /     2\    \4 + x /*sin(x)|                 4 + x                                                                                                                                   \4 + x /*sin (x)                                                                                                   sin (x)                                   /     2\                                                                             \4 + x /*sin(x)                                                                                                               /     2\                                                                                                                 \4 + x /*sin (x)                                                                                                                                                4 + x                                           \4 + x /*sin(x)               |            || sin(x)         2|                                                                                   /     2\                                                                           sin(x)                                        2               |     /  /     2\                    \
2    \                                 \                          \4 + x /                   /                                                                                                                                                                                                                                                                                                                      \4 + x /                                                                                                                                                                                                           \4 + x / *sin(x)                                                                                                                                                                                                                                                                                                                                                       /            \\           4 + x /                                                                                   \4 + x /*sin(x)                                                                                                             4 + x                /   3*\3*\4 + x /*cos(x) + 4*x*sin(x)/
-- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------
 3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    /     2\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  /     2\                                                                                                                                        2 /     2\               
x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     \4 + x /*sin(x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x*\4 + x /*sin(x)                                                                                                                                x *\4 + x /*sin(x)        
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    /     2\     3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    \4 + x / *sin (x)

\frac{- \frac{\left(\frac{40 x^{2} \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{\left(x^{2} + 4\right)^{2}} + 12 x \sin{\left(x \right)} + \frac{4 x \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \frac{48 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{12 x \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right)}{\left(x^{2} + 4\right)^{2} \sin{\left(x \right)}} + 9 \left(x^{2} + 4\right) \cos{\left(x \right)} + \frac{3 \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \left(\frac{24 x^{2}}{\left(x^{2} + 4\right)^{2}} + \frac{24 x \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} + 3 + \frac{12 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{4}{x^{2} + 4}\right) + \frac{21 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right)}{\left(x^{2} + 4\right) \sin{\left(x \right)}} - \frac{4 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} - \frac{9 \left(8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\left(x^{2} + 4\right) \sin^{2}{\left(x \right)}} + \frac{3 \left(24 x^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 12 x \left(x^{2} + 4\right) \sin^{3}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 8 x \sin^{3}{\left(x \right)} - 7 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 2 \left(x^{2} + 4\right)^{2} \cos^{3}{\left(x \right)} + 12 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)}{\left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}\right) \log{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{3 \left(\frac{4 x \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} + 4} + \left(\frac{4 x}{x^{2} + 4} + \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{8 x^{2} \sin^{2}{\left(x \right)} + 24 x \left(x^{2} + 4\right) \sin{\left(x \right)} \cos{\left(x \right)} - 3 \left(x^{2} + 4\right)^{2} \sin^{2}{\left(x \right)} + 6 \left(x^{2} + 4\right)^{2} \cos^{2}{\left(x \right)} + 4 \left(x^{2} + 4\right) \sin^{2}{\left(x \right)}}{\left(x^{2} + 4\right) \sin{\left(x \right)}}\right)}{x \left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{3 \left(4 x \sin{\left(x \right)} + 3 \left(x^{2} + 4\right) \cos{\left(x \right)}\right)}{x^{2} \left(x^{2} + 4\right) \sin{\left(x \right)}} + \frac{2}{x^{3}}}{\left(x^{2} + 4\right)^{2} \sin^{3}{\left(x \right)}}

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Derivada de y=(lnx)/((x²+4)²sin³x)

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Solución