Derivada de y=ln^2sinx/(lncosx)

Ha introducido [src]

   2          
log (x)*sin(x)
--------------
log(x)*cos(x)

\frac{\log{\left(x \right)}^{2} \sin{\left(x \right)}}{\log{\left(x \right)} \cos{\left(x \right)}}

(log(x)^2*sin(x))/((log(x)*cos(x)))

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de y=ln^2sinx/(lncosx)

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