Derivada de y=(sinx)^cosxtanx

Solución

Ha introducido [src]

   cos(x)          
sin      (x)*tan(x)

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

sin(x)^cos(x)*tan(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

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

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

Respuesta:

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

Gráfica

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Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

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Gráfico

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Derivada de y=(sinx)^cosxtanx

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