Derivada de y=sin^4(tg(x)/x)

Ha introducido [src]

   4/tan(x)\
sin |------|
    \  x   /

\sin^{4}{\left(\frac{\tan{\left(x \right)}}{x} \right)}

sin(tan(x)/x)^4

Solución detallada

Sustituimos $u = \sin{\left(\frac{\tan{\left(x \right)}}{x} \right)}$ .
Según el principio, aplicamos: $u^{4}$ tenemos $4 u^{3}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(\frac{\tan{\left(x \right)}}{x} \right)}$ :
1. Sustituimos $u = \frac{\tan{\left(x \right)}}{x}$ .
2. La derivada del seno es igual al coseno:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\tan{\left(x \right)}}{x}$ :
  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 $g{\left(x \right)} = x$ .
    
    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. Según el principio, aplicamos: $x$ tenemos $1$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}}{x^{2}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{\left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}\right) \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}}$
Como resultado de la secuencia de reglas:

$\frac{4 \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}\right) \sin^{3}{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}}$
Simplificamos:

$\frac{4 \left(\frac{x}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}\right) \sin^{3}{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}}$

Respuesta:

$\frac{4 \left(\frac{x}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}\right) \sin^{3}{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}}$

Gráfica

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Primera derivada [src]

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

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

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Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

  /                                                                                                                                                                                                                                                                /                                       2   \                                         /                                       2   \            \            
  |                                                                                                                                                    3                                        3                                 3/tan(x)\ /       2      tan(x)\ |tan(x)   /       2   \          1 + tan (x)|        2/tan(x)\ /       2      tan(x)\ |tan(x)   /       2   \          1 + tan (x)|    /tan(x)\|            
  |                                                                                                                              /       2      tan(x)\     3/tan(x)\     /       2      tan(x)\     2/tan(x)\    /tan(x)\   3*sin |------|*|1 + tan (x) - ------|*|------ + \1 + tan (x)/*tan(x) - -----------|   9*cos |------|*|1 + tan (x) - ------|*|------ + \1 + tan (x)/*tan(x) - -----------|*sin|------||            
  |             /             2                                          /       2   \     /       2   \       \               3*|1 + tan (x) - ------| *cos |------|   5*|1 + tan (x) - ------| *sin |------|*cos|------|         \  x   / \                x   / |   2                                 x     |         \  x   / \                x   / |   2                                 x     |    \  x   /|            
  |   2/tan(x)\ |/       2   \    3*tan(x)        2    /       2   \   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)|    /tan(x)\     \                x   /      \  x   /     \                x   /      \  x   /    \  x   /                                         \  x                                        /                                         \  x                                        /            |    /tan(x)\
8*|sin |------|*|\1 + tan (x)/  - -------- + 2*tan (x)*\1 + tan (x)/ + --------------- - ----------------------|*cos|------| + -------------------------------------- - -------------------------------------------------- - ----------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------|*sin|------|
  |    \  x   / |                     3                                        2                   x           |    \  x   /                      2                                              2                                                                    x                                                                                           x                                               |    \  x   /
  \             \                    x                                        x                                /                                 x                                              x                                                                                                                                                                                                                 /            
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                               x

\frac{8 \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{3 \tan{\left(x \right)}}{x^{3}}\right) \sin^{2}{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)} - \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x}\right) \sin^{3}{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x} + \frac{9 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x}\right) \sin{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos^{2}{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x}\right)^{3} \sin^{2}{\left(\frac{\tan{\left(x \right)}}{x} \right)} \cos{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x}\right)^{3} \cos^{3}{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x^{2}}\right) \sin{\left(\frac{\tan{\left(x \right)}}{x} \right)}}{x}

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Derivada de y=sin^4(tg(x)/x)

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