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Derivada de la función/ tan(x)/ x^exp/ x^exp(tan(x))*tan(x)

Derivada de x^exp(tan(x))*tan(x)

Solución

Ha introducido [src]

 / tan(x)\       
 \e      /       
x         *tan(x)

$$x^{e^{\tan{\left(x \right)}}} \tan{\left(x \right)}$$

x^exp(tan(x))*tan(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x^{e^{\tan{\left(x \right)}}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\tan{\left(x \right)} + 1\right) e^{e^{\tan{\left(x \right)}} \tan{\left(x \right)}}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de: $\frac{x^{e^{\tan{\left(x \right)}}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \left(\tan{\left(x \right)} + 1\right) e^{e^{\tan{\left(x \right)}} \tan{\left(x \right)}} \tan{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Primera derivada [src]

 / tan(x)\                  / tan(x)\ / tan(x)                               \       
 \e      / /       2   \    \e      / |e         /       2   \  tan(x)       |       
x         *\1 + tan (x)/ + x         *|------- + \1 + tan (x)/*e      *log(x)|*tan(x)
                                      \   x                                  /

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

Simplificar

Segunda derivada [src]

 / tan(x)\ /                         /                    2                                    2             /       2   \                                \                                                                    \
 \e      / |  /       2   \          |  1    /       2   \           /1   /       2   \       \   tan(x)   2*\1 + tan (x)/     /       2   \              |  tan(x)            /       2   \ /1   /       2   \       \  tan(x)|
x         *|2*\1 + tan (x)/*tan(x) + |- -- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e       + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e      *tan(x) + 2*\1 + tan (x)/*|- + \1 + tan (x)/*log(x)|*e      |
           |                         |   2                           \x                       /                   x                                       |                                  \x                       /        |
           \                         \  x                                                                                                                 /                                                                    /

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

Simplificar

Tercera derivada [src]

           /                                  /                                                                                                                                2                                                                                                                                                                                                                   \                                                                                                                                                                                                                            \
 / tan(x)\ |                                  |                  3                                    3               /       2   \                  2            /       2   \                                 /                    2            /       2   \                                \                                              /       2   \                         2              |                                  /                    2                                    2             /       2   \                                \                                                                    |
 \e      / |  /       2   \ /         2   \   |2    /       2   \           /1   /       2   \       \   2*tan(x)   3*\1 + tan (x)/     /       2   \           3*\1 + tan (x)/      /1   /       2   \       \ |  1    /       2   \           2*\1 + tan (x)/     /       2   \              |  tan(x)        2    /       2   \          6*\1 + tan (x)/*tan(x)     /       2   \               |  tan(x)            /       2   \ |  1    /       2   \           /1   /       2   \       \   tan(x)   2*\1 + tan (x)/     /       2   \              |  tan(x)     /       2   \ /1   /       2   \       \  tan(x)       |
x         *|2*\1 + tan (x)/*\1 + 3*tan (x)/ + |-- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e         - --------------- + 2*\1 + tan (x)/ *log(x) + ---------------- + 3*|- + \1 + tan (x)/*log(x)|*|- -- + \1 + tan (x)/ *log(x) + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e       + 4*tan (x)*\1 + tan (x)/*log(x) + ---------------------- + 6*\1 + tan (x)/ *log(x)*tan(x)|*e      *tan(x) + 3*\1 + tan (x)/*|- -- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e       + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e       + 6*\1 + tan (x)/*|- + \1 + tan (x)/*log(x)|*e      *tan(x)|
           |                                  | 3                           \x                       /                      2                                          x             \x                       / |   2                                  x                                       |                                                      x                                            |                                  |   2                           \x                       /                   x                                       |                           \x                       /               |
           \                                  \x                                                                           x                                                                                    \  x                                                                           /                                                                                                   /                                  \  x                                                                                                                 /                                                                    /

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

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Derivada de x^exp(tan(x))*tan(x)

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