Derivada de y=e^cos^x*xtgx

Ha introducido [src]

    x          
 cos (x)       
E       *tan(x)

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

E^(cos(x)^x)*tan(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = e^{\cos^{x}{\left(x \right)}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Sustituimos $u = \cos^{x}{\left(x \right)}$ .
2. Derivado $e^{u}$ es.
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos^{x}{\left(x \right)}$ :
  1. No logro encontrar los pasos en la búsqueda de esta derivada.
    
    Perola derivada
    
    $x^{x} \left(\log{\left(x \right)} + 1\right)$
  Como resultado de la secuencia de reglas:
  
  $x^{x} \left(\log{\left(x \right)} + 1\right) e^{\cos^{x}{\left(x \right)}}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $x^{x} \left(\log{\left(x \right)} + 1\right) e^{\cos^{x}{\left(x \right)}} \tan{\left(x \right)} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\cos^{x}{\left(x \right)}}}{\cos^{2}{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Primera derivada [src]

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

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

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Segunda derivada [src]

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

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

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Tercera derivada [src]

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

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

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Derivada de y=e^cos^x*xtgx

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