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Derivada de y=cos((x)^(2/x))

Función f() - derivada -er orden en el punto
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Solución

Ha introducido [src]
   / 2\
   | -|
   | x|
cos\x /
cos(x2x)\cos{\left(x^{\frac{2}{x}} \right)}
cos(x^(2/x))
Solución detallada
  1. Sustituimos u=x2xu = x^{\frac{2}{x}}.

  2. La derivada del coseno es igual a menos el seno:

    dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx2x\frac{d}{d x} x^{\frac{2}{x}}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

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

    Como resultado de la secuencia de reglas:

    (2x)2x(log(2x)+1)sin(x2x)- \left(\frac{2}{x}\right)^{\frac{2}{x}} \left(\log{\left(\frac{2}{x} \right)} + 1\right) \sin{\left(x^{\frac{2}{x}} \right)}


Respuesta:

(2x)2x(log(2x)+1)sin(x2x)- \left(\frac{2}{x}\right)^{\frac{2}{x}} \left(\log{\left(\frac{2}{x} \right)} + 1\right) \sin{\left(x^{\frac{2}{x}} \right)}

Primera derivada [src]
  2                    / 2\
  -                    | -|
  x /2    2*log(x)\    | x|
-x *|-- - --------|*sin\x /
    | 2       2   |        
    \x       x    /        
x2x(2log(x)x2+2x2)sin(x2x)- x^{\frac{2}{x}} \left(- \frac{2 \log{\left(x \right)}}{x^{2}} + \frac{2}{x^{2}}\right) \sin{\left(x^{\frac{2}{x}} \right)}
Segunda derivada [src]
      /                                              / 2\      2                   / 2\\
    2 |                   / 2\                       | -|      -                   | -||
    - |                   | -|                  2    | x|      x              2    | x||
    x |                   | x|   2*(-1 + log(x)) *sin\x /   2*x *(-1 + log(x)) *cos\x /|
-2*x *|(-3 + 2*log(x))*sin\x / + ------------------------ + ---------------------------|
      \                                     x                            x             /
----------------------------------------------------------------------------------------
                                            3                                           
                                           x                                            
2x2x((2log(x)3)sin(x2x)+2x2x(log(x)1)2cos(x2x)x+2(log(x)1)2sin(x2x)x)x3- \frac{2 x^{\frac{2}{x}} \left(\left(2 \log{\left(x \right)} - 3\right) \sin{\left(x^{\frac{2}{x}} \right)} + \frac{2 x^{\frac{2}{x}} \left(\log{\left(x \right)} - 1\right)^{2} \cos{\left(x^{\frac{2}{x}} \right)}}{x} + \frac{2 \left(\log{\left(x \right)} - 1\right)^{2} \sin{\left(x^{\frac{2}{x}} \right)}}{x}\right)}{x^{3}}
Tercera derivada [src]
     /                                               / 2\      4                   / 2\                                      / 2\       2                   / 2\      2                                  / 2\\
   2 |                    / 2\                       | -|      -                   | -|                                      | -|       -                   | -|      -                                  | -||
   - |                    | -|                  3    | x|      x              3    | x|                                      | x|       x              3    | x|      x                                  | x||
   x |                    | x|   4*(-1 + log(x)) *sin\x /   4*x *(-1 + log(x)) *sin\x /   6*(-1 + log(x))*(-3 + 2*log(x))*sin\x /   12*x *(-1 + log(x)) *cos\x /   6*x *(-1 + log(x))*(-3 + 2*log(x))*cos\x /|
2*x *|(-11 + 6*log(x))*sin\x / + ------------------------ - --------------------------- + --------------------------------------- + ---------------------------- + ------------------------------------------|
     |                                       2                            2                                  x                                    2                                    x                     |
     \                                      x                            x                                                                       x                                                           /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                       4                                                                                                      
                                                                                                      x                                                                                                       
2x2x((6log(x)11)sin(x2x)+6x2x(log(x)1)(2log(x)3)cos(x2x)x+6(log(x)1)(2log(x)3)sin(x2x)x4x4x(log(x)1)3sin(x2x)x2+12x2x(log(x)1)3cos(x2x)x2+4(log(x)1)3sin(x2x)x2)x4\frac{2 x^{\frac{2}{x}} \left(\left(6 \log{\left(x \right)} - 11\right) \sin{\left(x^{\frac{2}{x}} \right)} + \frac{6 x^{\frac{2}{x}} \left(\log{\left(x \right)} - 1\right) \left(2 \log{\left(x \right)} - 3\right) \cos{\left(x^{\frac{2}{x}} \right)}}{x} + \frac{6 \left(\log{\left(x \right)} - 1\right) \left(2 \log{\left(x \right)} - 3\right) \sin{\left(x^{\frac{2}{x}} \right)}}{x} - \frac{4 x^{\frac{4}{x}} \left(\log{\left(x \right)} - 1\right)^{3} \sin{\left(x^{\frac{2}{x}} \right)}}{x^{2}} + \frac{12 x^{\frac{2}{x}} \left(\log{\left(x \right)} - 1\right)^{3} \cos{\left(x^{\frac{2}{x}} \right)}}{x^{2}} + \frac{4 \left(\log{\left(x \right)} - 1\right)^{3} \sin{\left(x^{\frac{2}{x}} \right)}}{x^{2}}\right)}{x^{4}}