Derivada de (x×x×sin(x))/ln(x)

Solución

Ha introducido [src]

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

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

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

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = x^{2} \sin{\left(x \right)}$ 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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
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:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de (x×x×sin(x))/ln(x)

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