Derivada de sqrt((1-cosx)/(1+cosx))

Ha introducido [src]

    ____________
   / 1 - cos(x) 
  /  ---------- 
\/   1 + cos(x)

\sqrt{\frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}}

sqrt((1 - cos(x))/(1 + cos(x)))

Solución detallada

Sustituimos $u = \frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}$ .
Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 1 - \cos{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)} + 1$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. diferenciamos $1 - \cos{\left(x \right)}$ miembro por miembro:
    1. La derivada de una constante $1$ es igual a cero.
    2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      Entonces, como resultado: $\sin{\left(x \right)}$
    Como resultado de: $\sin{\left(x \right)}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $\cos{\left(x \right)} + 1$ miembro por miembro:
    1. La derivada de una constante $1$ es igual a cero.
    2. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Como resultado de: $- \sin{\left(x \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{\left(1 - \cos{\left(x \right)}\right) \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$
Como resultado de la secuencia de reglas:

$\frac{\left(1 - \cos{\left(x \right)}\right) \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{2 \sqrt{\frac{1 - \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}} \left(\cos{\left(x \right)} + 1\right)^{2}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

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Segunda derivada [src]

                      /                                                                                                                                                   2        \
                      |                                                                          2    /     -1 + cos(x)\      2    /     -1 + cos(x)\   /     -1 + cos(x)\     2   |
    _________________ |               2           2                                           sin (x)*|-1 + -----------|   sin (x)*|-1 + -----------|   |-1 + -----------| *sin (x)|
   / -(-1 + cos(x))   |  cos(x)    sin (x)     sin (x)*(-1 + cos(x))   (-1 + cos(x))*cos(x)           \      1 + cos(x)/           \      1 + cos(x)/   \      1 + cos(x)/         |
  /  --------------- *|- ------ - ---------- + --------------------- + -------------------- + -------------------------- - -------------------------- + ---------------------------|
\/      1 + cos(x)    |    2      1 + cos(x)                   2          2*(1 + cos(x))           2*(-1 + cos(x))               2*(1 + cos(x))               4*(-1 + cos(x))      |
                      \                            (1 + cos(x))                                                                                                                    /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    -1 + cos(x)

\frac{\sqrt{- \frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1}} \left(\frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right)^{2} \sin^{2}{\left(x \right)}}{4 \left(\cos{\left(x \right)} - 1\right)} - \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \sin^{2}{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \sin^{2}{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} + \frac{\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{\left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \frac{\cos{\left(x \right)}}{2} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{\cos{\left(x \right)} - 1}

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Tercera derivada [src]

                      /                   2                                   2                                   2                                   2                                                                                                                                                                                                                                                                                             /               2                                   2                 \                                                             \       
                      |              2*sin (x)    (-1 + cos(x))*cos(x)   2*sin (x)*(-1 + cos(x))             2*sin (x)    (-1 + cos(x))*cos(x)   2*sin (x)*(-1 + cos(x))                                                                                                                                                                                                           3                               2             /     -1 + cos(x)\ |          2*sin (x)    (-1 + cos(x))*cos(x)   2*sin (x)*(-1 + cos(x))|                                                    2        |       
                      |    -cos(x) - ---------- + -------------------- + -----------------------   -cos(x) - ---------- + -------------------- + -----------------------                                                    2    /     -1 + cos(x)\   /     -1 + cos(x)\                                                             /     -1 + cos(x)\          /     -1 + cos(x)\     2        /     -1 + cos(x)\     2      3*|-1 + -----------|*|-cos(x) - ---------- + -------------------- + -----------------------|      2    /     -1 + cos(x)\     /     -1 + cos(x)\     2   |       
    _________________ |              1 + cos(x)        1 + cos(x)                         2                  1 + cos(x)        1 + cos(x)                         2                            2                         sin (x)*|-1 + -----------|   |-1 + -----------|*cos(x)        2                                             |-1 + -----------|*cos(x)   |-1 + -----------| *sin (x)   3*|-1 + -----------| *sin (x)     \      1 + cos(x)/ |          1 + cos(x)        1 + cos(x)                         2     |   sin (x)*|-1 + -----------|   3*|-1 + -----------| *sin (x)|       
   / -(-1 + cos(x))   |1                                                      (1 + cos(x))                                                            (1 + cos(x))          3*cos(x)      3*sin (x)      -1 + cos(x)             \      1 + cos(x)/   \      1 + cos(x)/          3*sin (x)*(-1 + cos(x))   3*(-1 + cos(x))*cos(x)   \      1 + cos(x)/          \      1 + cos(x)/              \      1 + cos(x)/                                 \                                                   (1 + cos(x))      /           \      1 + cos(x)/     \      1 + cos(x)/         |       
  /  --------------- *|- + --------------------------------------------------------------------- - --------------------------------------------------------------------- - ---------- - ------------- - -------------- + -------------------------- + ------------------------- + ----------------------- + ---------------------- - ------------------------- + --------------------------- + ----------------------------- + -------------------------------------------------------------------------------------------- - -------------------------- - -----------------------------|*sin(x)
\/      1 + cos(x)    |2                                -1 + cos(x)                                                              1 + cos(x)                                1 + cos(x)               2   2*(1 + cos(x))                      2              2*(-1 + cos(x))                         3                        2              2*(1 + cos(x))                             2                              2                                               4*(-1 + cos(x))                                          (1 + cos(x))*(-1 + cos(x))    4*(1 + cos(x))*(-1 + cos(x))|       
                      \                                                                                                                                                                 (1 + cos(x))                           (-1 + cos(x))                                           (1 + cos(x))             (1 + cos(x))                                           8*(-1 + cos(x))                4*(-1 + cos(x))                                                                                                                                                                   /       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                          -1 + cos(x)

\frac{\sqrt{- \frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1}} \left(\frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right)^{3} \sin^{2}{\left(x \right)}}{8 \left(\cos{\left(x \right)} - 1\right)^{2}} - \frac{3 \left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right)^{2} \sin^{2}{\left(x \right)}}{4 \left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right)^{2} \sin^{2}{\left(x \right)}}{4 \left(\cos{\left(x \right)} - 1\right)^{2}} - \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \left(\frac{\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2 \left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \cos{\left(x \right)} - \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{4 \left(\cos{\left(x \right)} - 1\right)} + \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \cos{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} - \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} + \frac{\left(\frac{\cos{\left(x \right)} - 1}{\cos{\left(x \right)} + 1} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} - \frac{\cos{\left(x \right)} - 1}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{3 \left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{3}} + \frac{1}{2} - \frac{\frac{\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2 \left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \cos{\left(x \right)} - \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\cos{\left(x \right)} + 1} - \frac{3 \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\frac{\left(\cos{\left(x \right)} - 1\right) \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2 \left(\cos{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} - \cos{\left(x \right)} - \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1}

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Derivada de sqrt((1-cosx)/(1+cosx))

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