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Derivada de:
Derivada de x|x|
Derivada de x^-7
Derivada de f(x)=√x
Derivada de (cosx)^x
Expresiones idénticas
y= dos ^(arcsin1/x)*(tg^ cinco)x
y es igual a 2 en el grado (arc seno de 1 dividir por x) multiplicar por (tg en el grado 5)x
y es igual a dos en el grado (arc seno de 1 dividir por x) multiplicar por (tg en el grado cinco)x
y=2(arcsin1/x)*(tg5)x
y=2arcsin1/x*tg5x
y=2^(arcsin1/x)*(tg⁵)x
y=2^(arcsin1/x)(tg^5)x
y=2(arcsin1/x)(tg5)x
y=2arcsin1/xtg5x
y=2^arcsin1/xtg^5x
y=2^(arcsin1 dividir por x)*(tg^5)x
Expresiones con funciones
tg
tg(cosx)

Derivada de la función/ y=2^(arcsin1/x)/ sin1/x/ arcsin/ y=2^(arcsin1/x)*(tg^5)x

Derivada de y=2^(arcsin1/x)*(tg^5)x

Solución

Ha introducido [src]

 asin(1)          
 -------          
    x       5     
2       *tan (x)*x

$$x 2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \tan^{5}{\left(x \right)}$$

(2^(asin(1)/x)*tan(x)^5)*x

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = 2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \tan^{5}{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Sustituimos $u = \frac{\operatorname{asin}{\left(1 \right)}}{x}$ .
  2. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\operatorname{asin}{\left(1 \right)}}{x}$ :
    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Según el principio, aplicamos: $\frac{1}{x}$ tenemos $- \frac{1}{x^{2}}$
      Entonces, como resultado: $- \frac{\operatorname{asin}{\left(1 \right)}}{x^{2}}$
    Como resultado de la secuencia de reglas:
    
    $- \frac{2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \log{\left(2 \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}}$
  $g{\left(x \right)} = \tan^{5}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Sustituimos $u = \tan{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{5}$ tenemos $5 u^{4}$
  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:
    
    $\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de: $\frac{5 \cdot 2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \log{\left(2 \right)} \tan^{5}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}}$
$g{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
Como resultado de: $2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \tan^{5}{\left(x \right)} + x \left(\frac{5 \cdot 2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{4}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \log{\left(2 \right)} \tan^{5}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}}\right)$
Simplificamos:

$\frac{2^{-1 + \frac{\pi}{2 x}} \left(10 x^{2} + x \sin{\left(2 x \right)} - \frac{\pi \log{\left(2 \right)} \sin{\left(2 x \right)}}{2}\right) \tan^{4}{\left(x \right)}}{x \cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{2^{-1 + \frac{\pi}{2 x}} \left(10 x^{2} + x \sin{\left(2 x \right)} - \frac{\pi \log{\left(2 \right)} \sin{\left(2 x \right)}}{2}\right) \tan^{4}{\left(x \right)}}{x \cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

  /                                    asin(1)                       \                   
  | asin(1)                            -------                       |    asin(1)        
  | -------                               x       5                  |    -------        
  |    x       4    /         2   \   2       *tan (x)*asin(1)*log(2)|       x       5   
x*|2       *tan (x)*\5 + 5*tan (x)/ - -------------------------------| + 2       *tan (x)
  |                                                   2              |                   
  \                                                  x               /

$$2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \tan^{5}{\left(x \right)} + x \left(2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \left(5 \tan^{2}{\left(x \right)} + 5\right) \tan^{4}{\left(x \right)} - \frac{2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \log{\left(2 \right)} \tan^{5}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}}\right)$$

Simplificar

Segunda derivada [src]

 asin(1)         /  /                                      2    /    asin(1)*log(2)\                                                        \                                                     \
 -------         |  |                                   tan (x)*|2 + --------------|*asin(1)*log(2)      /       2   \                      |                                  2                  |
    x       3    |  |   /       2   \ /         2   \           \          x       /                  10*\1 + tan (x)/*asin(1)*log(2)*tan(x)|      /       2   \          2*tan (x)*asin(1)*log(2)|
2       *tan (x)*|x*|10*\1 + tan (x)/*\2 + 3*tan (x)/ + ------------------------------------------- - --------------------------------------| + 10*\1 + tan (x)/*tan(x) - ------------------------|
                 |  |                                                         3                                          2                  |                                         2           |
                 \  \                                                        x                                          x                   /                                        x            /

$$2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \left(x \left(10 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) - \frac{10 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}} + \frac{\left(2 + \frac{\log{\left(2 \right)} \operatorname{asin}{\left(1 \right)}}{x}\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{3}}\right) + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \log{\left(2 \right)} \tan^{2}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}}\right) \tan^{3}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

                 /  /                                                                                     /        2       2                      \                                                                                                                                       \                                                                                                                                     \
                 |  |                                                                                3    |    asin (1)*log (2)   6*asin(1)*log(2)|                                                                                                                                       |                                                                                                                                     |
 asin(1)         |  |                                                                             tan (x)*|6 + ---------------- + ----------------|*asin(1)*log(2)                                                                  2    /       2   \ /    asin(1)*log(2)\               |     /                                      2    /    asin(1)*log(2)\                                                        \       |
 -------         |  |                 /                           2                           \           |            2                 x        |                     /       2   \ /         2   \                         15*tan (x)*\1 + tan (x)/*|2 + --------------|*asin(1)*log(2)|     |                                   tan (x)*|2 + --------------|*asin(1)*log(2)      /       2   \                      |       |
    x       2    |  |   /       2   \ |     4        /       2   \          2    /       2   \|           \           x                           /                  30*\1 + tan (x)/*\2 + 3*tan (x)/*asin(1)*log(2)*tan(x)                            \          x       /               |     |   /       2   \ /         2   \           \          x       /                  10*\1 + tan (x)/*asin(1)*log(2)*tan(x)|       |
2       *tan (x)*|x*|10*\1 + tan (x)/*\2*tan (x) + 6*\1 + tan (x)/  + 13*tan (x)*\1 + tan (x)// - ---------------------------------------------------------------- - ------------------------------------------------------ + ------------------------------------------------------------| + 3*|10*\1 + tan (x)/*\2 + 3*tan (x)/ + ------------------------------------------- - --------------------------------------|*tan(x)|
                 |  |                                                                                                             4                                                             2                                                           3                             |     |                                                         3                                          2                  |       |
                 \  \                                                                                                            x                                                             x                                                           x                              /     \                                                        x                                          x                   /       /

$$2^{\frac{\operatorname{asin}{\left(1 \right)}}{x}} \left(x \left(10 \left(\tan^{2}{\left(x \right)} + 1\right) \left(6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 13 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) - \frac{30 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) \log{\left(2 \right)} \tan{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}} + \frac{15 \left(2 + \frac{\log{\left(2 \right)} \operatorname{asin}{\left(1 \right)}}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{3}} - \frac{\left(6 + \frac{6 \log{\left(2 \right)} \operatorname{asin}{\left(1 \right)}}{x} + \frac{\log{\left(2 \right)}^{2} \operatorname{asin}^{2}{\left(1 \right)}}{x^{2}}\right) \log{\left(2 \right)} \tan^{3}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{4}}\right) + 3 \left(10 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 2\right) - \frac{10 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{2}} + \frac{\left(2 + \frac{\log{\left(2 \right)} \operatorname{asin}{\left(1 \right)}}{x}\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)} \operatorname{asin}{\left(1 \right)}}{x^{3}}\right) \tan{\left(x \right)}\right) \tan^{2}{\left(x \right)}$$

Simplificar

Gráfico

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Derivada de y=2^(arcsin1/x)*(tg^5)x

Gráfico:

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Solución