Derivada de x*sinx/lnx

Solución

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x \sin{\left(x \right)}$ 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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
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:

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

Respuesta:

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

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

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