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y=√((2cos^2x-1)÷(cosx-senx))

Derivada de y=√((2cos^2x-1)÷(cosx-senx))

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
     _________________
    /       2         
   /   2*cos (x) - 1  
  /   --------------- 
\/    cos(x) - sin(x) 
$$\sqrt{\frac{2 \cos^{2}{\left(x \right)} - 1}{- \sin{\left(x \right)} + \cos{\left(x \right)}}}$$
sqrt((2*cos(x)^2 - 1)/(cos(x) - sin(x)))
Solución detallada
  1. Sustituimos .

  2. Según el principio, aplicamos: tenemos

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

    1. Se aplica la regla de la derivada parcial:

      y .

      Para calcular :

      1. diferenciamos miembro por miembro:

        1. La derivada de una constante 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. Sustituimos .

          2. Según el principio, aplicamos: tenemos

          3. Luego se aplica una cadena de reglas. Multiplicamos por :

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

            Como resultado de la secuencia de reglas:

          Entonces, como resultado:

        Como resultado de:

      Para calcular :

      1. diferenciamos miembro por miembro:

        1. 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 seno es igual al coseno:

          Entonces, como resultado:

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

        Como resultado de:

      Ahora aplicamos la regla de la derivada de una divesión:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Primera derivada [src]
     _________________                                                                        
    /       2          //     2       \                                    \                  
   /   2*cos (x) - 1   |\2*cos (x) - 1/*(cos(x) + sin(x))   2*cos(x)*sin(x)|                  
  /   --------------- *|--------------------------------- - ---------------|*(cos(x) - sin(x))
\/    cos(x) - sin(x)  |                          2         cos(x) - sin(x)|                  
                       \       2*(cos(x) - sin(x))                         /                  
----------------------------------------------------------------------------------------------
                                             2                                                
                                        2*cos (x) - 1                                         
$$\frac{\sqrt{\frac{2 \cos^{2}{\left(x \right)} - 1}{- \sin{\left(x \right)} + \cos{\left(x \right)}}} \left(- \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{2 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\right) \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \cos^{2}{\left(x \right)} - 1}$$
Segunda derivada [src]
                          /                                                                                  2                                                                                                                                                                                                                              \
                          |                            /                  /          2   \                  \                                          /                  /          2   \                  \                       /                  /          2   \                  \                                                  |
     ____________________ |                            |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                          |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                       |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                                  |
    /  /          2   \   |                            |4*cos(x)*sin(x) + ----------------------------------|                     2 /          2   \   |4*cos(x)*sin(x) + ----------------------------------|*(cos(x) + sin(x))   2*|4*cos(x)*sin(x) + ----------------------------------|*cos(x)*sin(x)                                    |
   /  -\-1 + 2*cos (x)/   |  1      2           2      \                           -cos(x) + sin(x)         /    (cos(x) + sin(x)) *\-1 + 2*cos (x)/   \                           -cos(x) + sin(x)         /                       \                           -cos(x) + sin(x)         /                 4*(cos(x) + sin(x))*cos(x)*sin(x)|
  /   ------------------ *|- - - cos (x) + 2*sin (x) + ------------------------------------------------------- + ----------------------------------- - ------------------------------------------------------------------------ - ---------------------------------------------------------------------- + ---------------------------------|
\/     -cos(x) + sin(x)   |  2                                              /          2   \                                               2                                     2*(-cos(x) + sin(x))                                                                   2                                           -cos(x) + sin(x)        |
                          \                                               4*\-1 + 2*cos (x)/                             (-cos(x) + sin(x))                                                                                                                   -1 + 2*cos (x)                                                                /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                          2                                                                                                                                                                  
                                                                                                                                                                -1 + 2*cos (x)                                                                                                                                                               
$$\frac{\sqrt{- \frac{2 \cos^{2}{\left(x \right)} - 1}{\sin{\left(x \right)} - \cos{\left(x \right)}}} \left(\frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2}}{4 \left(2 \cos^{2}{\left(x \right)} - 1\right)} - \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)} + 2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} - \frac{1}{2} + \frac{4 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{2 \cos^{2}{\left(x \right)} - 1}$$
Tercera derivada [src]
                          /                                                                        3                     /                                                2 /          2   \                                    \                                                                                                                                                                                                                                                                                      /                  /          2   \                  \ /                                                2 /          2   \                                    \                                                                                                                                                                             2                   /                                                2 /          2   \                                    \                                                                         2                                                                                                             \
                          |                  /                  /          2   \                  \                      |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                                                                                   /                  /          2   \                  \                                                    /                  /          2   \                  \                                   |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))| |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                      /                  /          2   \                  \                                           /                  /          2   \                  \                    |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                   /                  /          2   \                  \                        /                  /          2   \                  \                                |
     ____________________ |                  |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|    (cos(x) + sin(x))*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|                                                                              2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                               2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                 3*|4*cos(x)*sin(x) + ----------------------------------|*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|         2       2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                           |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                  4*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|*cos(x)*sin(x)     |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                        |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                |
    /  /          2   \   |                  |4*cos(x)*sin(x) + ----------------------------------|                      |                                                        2                     -cos(x) + sin(x)        |        2                                           3 /          2   \   2*cos (x)*|4*cos(x)*sin(x) + ----------------------------------|     /          2   \                     2*sin (x)*|4*cos(x)*sin(x) + ----------------------------------|        2                          \                           -cos(x) + sin(x)         / |                                                        2                     -cos(x) + sin(x)        |   16*cos (x)*sin (x)*|4*cos(x)*sin(x) + ----------------------------------|                       2                 3*|4*cos(x)*sin(x) + ----------------------------------| *cos(x)*sin(x)     |                                                        2                     -cos(x) + sin(x)        |                 3*|4*cos(x)*sin(x) + ----------------------------------| *(cos(x) + sin(x))   4*|4*cos(x)*sin(x) + ----------------------------------|*(cos(x) + sin(x))*cos(x)*sin(x)|
   /  -\-1 + 2*cos (x)/   |                  \                           -cos(x) + sin(x)         /                      \                                      (-cos(x) + sin(x))                                              /   6*sin (x)*(cos(x) + sin(x))   3*(cos(x) + sin(x)) *\-1 + 2*cos (x)/             \                           -cos(x) + sin(x)         /   2*\-1 + 2*cos (x)/*(cos(x) + sin(x))             \                           -cos(x) + sin(x)         /   6*cos (x)*(cos(x) + sin(x))                                                            \                                      (-cos(x) + sin(x))                                              /                      \                           -cos(x) + sin(x)         /   12*(cos(x) + sin(x)) *cos(x)*sin(x)     \                           -cos(x) + sin(x)         /                    \                                      (-cos(x) + sin(x))                                              /                   \                           -cos(x) + sin(x)         /                        \                           -cos(x) + sin(x)         /                                |
  /   ------------------ *|4*cos(x)*sin(x) - ------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------- - --------------------------- - ------------------------------------- - ---------------------------------------------------------------- - ------------------------------------ + ---------------------------------------------------------------- + --------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------- - ----------------------------------- + ----------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|
\/     -cos(x) + sin(x)   |                                                      2                                                                          -cos(x) + sin(x)                                                              -cos(x) + sin(x)                                   3                                               2                                         -cos(x) + sin(x)                                                2                                     -cos(x) + sin(x)                                                                                  /          2   \                                                                                                                      2                                                         2                                                      2                                                                                             2                                                                                /          2   \                                                                   /          2   \                                              |
                          |                                      /          2   \                                                                                                                                                                                          (-cos(x) + sin(x))                                      -1 + 2*cos (x)                                                                                            -1 + 2*cos (x)                                                                                                                                  4*\-1 + 2*cos (x)/                                                                                                      /          2   \                                        (-cos(x) + sin(x))                                       /          2   \                                                                                    -1 + 2*cos (x)                                                                           4*\-1 + 2*cos (x)/*(-cos(x) + sin(x))                                                \-1 + 2*cos (x)/*(-cos(x) + sin(x))                           |
                          \                                    8*\-1 + 2*cos (x)/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    \-1 + 2*cos (x)/                                                                                                 \-1 + 2*cos (x)/                                                                                                                                                                                                                                                                                                                                /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     -1 + 2*cos (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
$$\frac{\sqrt{- \frac{2 \cos^{2}{\left(x \right)} - 1}{\sin{\left(x \right)} - \cos{\left(x \right)}}} \left(- \frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{3}}{8 \left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{4 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)} - \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{4 \left(2 \cos^{2}{\left(x \right)} - 1\right)} + \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{16 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} - \frac{4 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)} + 4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{4 \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}} - \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} - \frac{12 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}} - \frac{3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{3} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{3}}\right)}{2 \cos^{2}{\left(x \right)} - 1}$$
Gráfico
Derivada de y=√((2cos^2x-1)÷(cosx-senx))