Derivada de y=2^cos(tgx)

Solución

Ha introducido [src]

 cos(tan(x))
2

$$2^{\cos{\left(\tan{\left(x \right)} \right)}}$$

2^cos(tan(x))

Solución detallada

Sustituimos $u = \cos{\left(\tan{\left(x \right)} \right)}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(\tan{\left(x \right)} \right)}$ :
1. Sustituimos $u = \tan{\left(x \right)}$ .
2. La derivada del coseno es igual a menos el seno:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
  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 la secuencia de reglas:
  
  $- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\tan{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

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

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

Respuesta:

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

Gráfica

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Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de y=2^cos(tgx)

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