Derivada de -xsin(x)+tg(x)cos(x)*ctg(x)

Solución

Ha introducido [src]

-x*sin(x) + tan(x)*cos(x)*cot(x)

$$- x \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)}$$

(-x)*sin(x) + (tan(x)*cos(x))*cot(x)

Solución detallada

diferenciamos $- x \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} \cot{\left(x \right)}$ miembro por miembro:
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. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
    1. Según el principio, aplicamos: $x$ tenemos $1$
    Entonces, como resultado: $-1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $- x \cos{\left(x \right)} - \sin{\left(x \right)}$
2. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(x \right)} \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Reescribimos las funciones para diferenciar:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
    2. 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:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    $g{\left(x \right)} = \cos{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Como resultado de: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}$
  $g{\left(x \right)} = \cot{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Hay varias formas de calcular esta derivada.
    Method #1
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
    2. Sustituimos $u = \tan{\left(x \right)}$ .
    3. Según el principio, aplicamos: $\frac{1}{u}$ tenemos $- \frac{1}{u^{2}}$
    4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      
      Como resultado de la secuencia de reglas:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
    Method #2
    1. Reescribimos las funciones para diferenciar:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
    2. Se aplica la regla de la derivada parcial:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
      
      La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
      
      La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Ahora aplicamos la regla de la derivada de una divesión:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  Como resultado de: $\left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \cot{\left(x \right)} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}}$
Como resultado de: $- x \cos{\left(x \right)} + \left(\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan{\left(x \right)}\right) \cot{\left(x \right)} - \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} - \sin{\left(x \right)}$
Simplificamos:

$- x \cos{\left(x \right)} - 2 \sin{\left(x \right)}$

Respuesta:

$- x \cos{\left(x \right)} - 2 \sin{\left(x \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

Derivada de -x*sin(x)+tg(x)*cos(x)*ctg(x)

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Expresiones con funciones

Derivada de -xsin(x)+tg(x)cos(x)*ctg(x)

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Derivada de -x*sin(x)+tg(x)*cos(x)*ctg(x)

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2

Derivada de -xsin(x)+tg(x)cos(x)*ctg(x)