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Expresiones idénticas
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Expresiones semejantes
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Derivada de la función/ ln^2x/ xln^2/ xsinx/ (xsinx)/(1+tgx)/ (xsinx)/(1+tgx)+xln^2x

Derivada de (xsinx)/(1+tgx)+xln^2x

Solución

Ha introducido [src]

 x*sin(x)         2   
---------- + x*log (x)
1 + tan(x)

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

(x*sin(x))/(1 + tan(x)) + x*log(x)^2

Solución detallada

diferenciamos $x \log{\left(x \right)}^{2} + \frac{x \sin{\left(x \right)}}{\tan{\left(x \right)} + 1}$ miembro por miembro:
1. 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 $g{\left(x \right)} = \tan{\left(x \right)} + 1$ .
  
  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. diferenciamos $\tan{\left(x \right)} + 1$ miembro por miembro:
    1. La derivada de una constante $1$ es igual a cero.
    2. Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    3. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      1. La derivada del seno es igual al coseno:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    Como resultado de: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}}$
2. 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 $1$
  $g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  Como resultado de: $\log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$
Como resultado de: $\frac{- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                           2                                                                               2                                                    3                                                           2                                                                        2                                                                                                 2              
                                     /       2   \            /       2   \                          /       2   \                            /       2   \             /       2   \              /       2   \             /       2   \                     /       2   \             /       2   \                                  /       2   \                  2    /       2   \              /       2   \                      /       2   \               
  3*sin(x) + x*cos(x)   2*log(x)   2*\1 + tan (x)/*cos(x)   2*\1 + tan (x)/ *(x*cos(x) + sin(x))   2*\1 + tan (x)/*(-2*cos(x) + x*sin(x))   4*\1 + tan (x)/ *sin(x)   x*\1 + tan (x)/*sin(x)   6*x*\1 + tan (x)/ *sin(x)   4*\1 + tan (x)/*sin(x)*tan(x)   2*x*\1 + tan (x)/ *sin(x)   2*\1 + tan (x)/*(x*cos(x) + sin(x))*tan(x)   4*x*\1 + tan (x)/ *cos(x)   4*x*tan (x)*\1 + tan (x)/*sin(x)   4*x*\1 + tan (x)/*cos(x)*tan(x)   12*x*\1 + tan (x)/ *sin(x)*tan(x)
- ------------------- - -------- - ---------------------- + ------------------------------------ + -------------------------------------- + ----------------------- + ---------------------- - ------------------------- - ----------------------------- - ------------------------- - ------------------------------------------ + ------------------------- - -------------------------------- - ------------------------------- + ---------------------------------
       1 + tan(x)           2                      2                               3                                       2                                 3                        2                          4                             2                             2                                   2                                    3                              2                                  2                                  3          
                           x           (1 + tan(x))                    (1 + tan(x))                            (1 + tan(x))                      (1 + tan(x))             (1 + tan(x))               (1 + tan(x))                  (1 + tan(x))                  (1 + tan(x))                        (1 + tan(x))                         (1 + tan(x))                   (1 + tan(x))                       (1 + tan(x))                       (1 + tan(x))

$$- \frac{2 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{12 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} + \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{6 x \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{4}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{x \cos{\left(x \right)} + 3 \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{2 \log{\left(x \right)}}{x^{2}}$$

Simplificar

Gráfico

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de (xsinx)/(1+tgx)+xln^2x

Gráfico:

Definida a trozos:

Solución