Derivada de y=lnsin3x/lnsinx

Ha introducido [src]

log(sin(3*x))       
-------------*sin(x)
    log(x)

\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}} \sin{\left(x \right)}

(log(sin(3*x))/log(x))*sin(x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

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

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

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

  /                  /      2   \                                  \                                                                      
  |         2        |1 + ------|*log(sin(3*x))                    |                                                                      
  |    9*cos (3*x)   \    log(x)/                     6*cos(3*x)   |                                   /3*cos(3*x)   log(sin(3*x))\       
- |9 + ----------- - -------------------------- + -----------------|*sin(x) - log(sin(3*x))*sin(x) + 2*|---------- - -------------|*cos(x)
  |        2                  2                   x*log(x)*sin(3*x)|                                   \ sin(3*x)       x*log(x)  /       
  \     sin (3*x)            x *log(x)                             /                                                                      
------------------------------------------------------------------------------------------------------------------------------------------
                                                                  log(x)

\frac{2 \left(\frac{3 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{x \log{\left(x \right)}}\right) \cos{\left(x \right)} - \left(9 + \frac{9 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{6 \cos{\left(3 x \right)}}{x \log{\left(x \right)} \sin{\left(3 x \right)}} - \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}}{x^{2} \log{\left(x \right)}}\right) \sin{\left(x \right)} - \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(x \right)}}{\log{\left(x \right)}}

Simplificar

Tercera derivada [src]

/   /       2     \      /       2     \                                                                            \                                                                                                                                                    
|   |    cos (3*x)|      |    cos (3*x)|              /      3         3   \                                        |                                                                                                                                                    
|27*|1 + ---------|   54*|1 + ---------|*cos(3*x)   2*|1 + ------ + -------|*log(sin(3*x))     /      2   \         |                                                                           /                  /      2   \                                  \       
|   |       2     |      |       2     |              |    log(x)      2   |                 9*|1 + ------|*cos(3*x)|                                                                           |         2        |1 + ------|*log(sin(3*x))                    |       
|   \    sin (3*x)/      \    sin (3*x)/              \             log (x)/                   \    log(x)/         |                                   /3*cos(3*x)   log(sin(3*x))\            |    9*cos (3*x)   \    log(x)/                     6*cos(3*x)   |       
|------------------ + --------------------------- - -------------------------------------- + -----------------------|*sin(x) - cos(x)*log(sin(3*x)) - 3*|---------- - -------------|*sin(x) - 3*|9 + ----------- - -------------------------- + -----------------|*cos(x)
|     x*log(x)                  sin(3*x)                           3                             2                  |                                   \ sin(3*x)       x*log(x)  /            |        2                  2                   x*log(x)*sin(3*x)|       
\                                                                 x *log(x)                     x *log(x)*sin(3*x)  /                                                                           \     sin (3*x)            x *log(x)                             /       
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                  log(x)

\frac{- 3 \left(\frac{3 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}}{x \log{\left(x \right)}}\right) \sin{\left(x \right)} - 3 \left(9 + \frac{9 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} + \frac{6 \cos{\left(3 x \right)}}{x \log{\left(x \right)} \sin{\left(3 x \right)}} - \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}}{x^{2} \log{\left(x \right)}}\right) \cos{\left(x \right)} + \left(\frac{54 \left(1 + \frac{\cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} + \frac{27 \left(1 + \frac{\cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right)}{x \log{\left(x \right)}} + \frac{9 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)} \sin{\left(3 x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \log{\left(\sin{\left(3 x \right)} \right)}}{x^{3} \log{\left(x \right)}}\right) \sin{\left(x \right)} - \log{\left(\sin{\left(3 x \right)} \right)} \cos{\left(x \right)}}{\log{\left(x \right)}}

Simplificar

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