Derivada de y=tgx/cosx+ctgx/sinx

Solución

Ha introducido [src]

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

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

tan(x)/cos(x) + cot(x)/sin(x)

Solución detallada

diferenciamos $\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\cot{\left(x \right)}}{\sin{\left(x \right)}}$ miembro por miembro:
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \tan{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  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:
  
  $\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} + \sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
2. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cot{\left(x \right)}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cos{\left(x \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
Como resultado de: $\frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} + \sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cos{\left(x \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                   2                  2                                                                                                                                                                                                                                                                    
    /       2   \     /       2   \      /       2   \      /       2   \        2    /       2   \        3                                    2    /       2   \        2    /       2   \                          3                  2    /       2   \     /       2   \                   /       2   \              
  3*\1 + cot (x)/   2*\1 + cot (x)/    2*\1 + tan (x)/    3*\1 + tan (x)/   6*cos (x)*\1 + cot (x)/   6*cos (x)*cot(x)   5*cos(x)*cot(x)   4*cot (x)*\1 + cot (x)/   4*tan (x)*\1 + tan (x)/   5*sin(x)*tan(x)   6*sin (x)*tan(x)   6*sin (x)*\1 + tan (x)/   6*\1 + cot (x)/*cos(x)*cot(x)   6*\1 + tan (x)/*sin(x)*tan(x)
- --------------- - ---------------- + ---------------- + --------------- - ----------------------- - ---------------- - --------------- - ----------------------- + ----------------------- + --------------- + ---------------- + ----------------------- - ----------------------------- + -----------------------------
       sin(x)            sin(x)             cos(x)             cos(x)                  3                     4                  2                   sin(x)                    cos(x)                  2                 4                      3                            2                               2              
                                                                                    sin (x)               sin (x)            sin (x)                                                               cos (x)           cos (x)                cos (x)                      sin (x)                         cos (x)

$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\cos{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\cos{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\sin{\left(x \right)}} - \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}}{\sin{\left(x \right)}} - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)}{\sin{\left(x \right)}} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{6 \sin^{3}{\left(x \right)} \tan{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{5 \sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{5 \cos{\left(x \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \cos^{3}{\left(x \right)} \cot{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$

Simplificar

Gráfico

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Derivada de y=tgx/cosx+ctgx/sinx

Gráfico:

Definida a trozos:

Solución

Method #1

Method #2