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

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

Solución

Ha introducido [src]

 -tan(x)  
x       *x

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

x^(-tan(x))*x

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ y $g{\left(x \right)} = x^{\tan{\left(x \right)}}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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

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